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LIBRARY OF CONGRESS.! 



UNITED STATES OP AMERICA. 






The Metal Worker 



PATTERN BOOK. 



A PRACTICAL TREATISE ON THE ART AND SCIENCE OF PATTERN 
CUTTING AS APPLIED TO SHEET METAL WORK. 



yy 



- 












BY 



A. O. KITTREDGE. 



d 



New Yoek: 
david williams, 83 reade street. 

1882. 




Copyright, 18S1, 
By DAVID WILLIAME 



Tress and Bindery of 

S. W. GREEN'S SON, 

74 & 76 Beekman St., 

NEW YORK. 



PREFACE. 



The demand for a second edition of the "Metal Worker Pattern Book," the author feels, may he taken 
as conclusive evidence that his work has been found useful by those for whom it was specially intended. It 
was undertaken in response to a well-defined demand upon the part of Tinners and Cornice-makers for a 
comprehensive exposition of the principles of pattern cutting as applied to sheet-metal work. Some parts were 
prepared in direct answer to questions asked by subscribers of The Metal Worker, and appeared in the 
columns of that journal during the time the work was in progress. The most careful attention has been given 
throughout to the needs of sheet-metal workers, as made known to the author during his editorial connection 
with The Metal Worker, and thnragh his previous practical experience in the trade. The aim has been to 
present a work serviceable alike to the apprentice boy who can afford but a single instruction book, to the 
mechanic who desires to add to whatever knowledge of pattern cutting he already possesses, to the student who 
would master the art by systematic investigation, and to all who need occasional assistance in getting over 
difficult places. 

The work is comprised in five general divisions or chapters. In the first, Definitions and Technicalities 
are considered. The various mathematical and mechanical terms which it has been found necessary to use in 
the book, and which are current among mechanics, are explained, and such architectural terms as are commonly 
employed in cornice shops have been defined. Illustrations have been employed wherever their use has 
seemed advantageous. Following this is a chapter on Drawing Tools and Materials, prepared to meet the 
wants of those who commence drawing as a preliminary step to pattern cutting. This in turn is followed by a 
selection of simple Geometrical Problems, chosen with particular reference to the needs of students of pattern 
cutting. Various ways of solving the same problem, and the use of different instruments in accomplishing the 
same object, are presented in order to give the mechanic the widest range of choice in methods. At this stage 
the theoretical chapter of the work is presented, and is entitled the "Art and Science of Pattern Cutting." It 
is an attempt to explain the principles underlying all the operations of pattern cutting in such a way as will 
enable the student to make intelligent application of them, irrespective of formulated rules. Mechanics who 
already possess a fair degree of ability as pattern cutters, but who are perplexed when unusual combinations 
arise, may find in this division of the book all they require to render them proficient. Following this is a 
selection of Pattern Problems, arranged for the most part in classes, so that those of a kind will be found 
together. Each problem, so far as possible, has been demonstrated independently of all others. This 
arrangement has been followed to facilitate occasional use of the book by those who do not care to go through 
it in course. The- work is completed by a comprehensive index, which it is believed will be found useful when 
searching for any required problem. The names of some of the ordinary articles of ware made in tin shops 
have been included in it, with references to the rules which may be employed in developing their patterns. 

The " Metal "Worker Pattern Book " was prepared from the mechanic's standpoint. In diction and 
style it will be found suited to the needs of workmen of the most ordinary attainments. Each proposition is 
expressed and demonstrated in language which the average reader will have no difficulty in understanding, and 
which the apprentice can read without becoming confused or discouraged. 

83 Reade Street, New Yoke, November, 18S2. 



Digitized by the Internet Archive 
in 2011 with funding from 
The Library of Congress 



http://www.archive.org/details/metalworkerpatteOOkitt 



DEFINITIONS AND TECHNICALITIES. 



j 1. A treatise on Pattern Cutting as applied to sheet-metal work, is only an exposition and application of 
geometrical principles. Any work on geometry, or, more particularly, upon geometrical drawing, presents in 
a general way all the principles that enter into the art of Pattern Cutting. It remains for us, therefore in 
this work to make specific application of those principles, and to describe them in a way that will be readily 
understood by mechanies who have not had the advantages of a mathematical education. "While in each 
problem and demonstration we shall be careful to avoid, as far as possible, the use of technical terms and words 
not in common use among mechanics, the necessity for precise language in describing geometrical fi<mres to- 
gether with the fact that the every-day vocabulary of the workshop is not 
F . g ^_ a stra . ght Line _ sufficiently comprehensive to enable us to restrict ourselves entirely to it, 

compels us to employ some terms not in general use which it is proper 
we should define and explain at the outset. In this connection it may not be out of place to remark that 
the advantage to the mechanic of accurate language is so considerable that every student of this book will be 
justified in giving careful attention to the introductory chapter, for the purpose of increasing and improving 
his vocabulary, as well as for the sake of being able to readily comprehend the demonstrations in the pao-es fol- 
lowing. For this reason we have extended the list of terms to be defined somewhat beyond the strict requirements 
of this book. We have made it include nearly all of the terms belonging to plane geometry and those pecu- 
liar to pattern cutting, and we have added a few architectural terms suggested by the problems relating to 
cornice work. "We introduce the terms and definitions in the way of a familiar talk, rather than in the set form 
of a glossary, because we believe the former will be of greater advantage to the mechanic. By reference to the 
index, which is arranged alphabetically, any term can be readily found. 

2. Geometry is that branch of mathematics which investigates the rela- 
tions, properties and measurements of solids, surfaces, lines and angles. 

3. Sheet-Metal Pattern Cutting is founded upon those principles of ge- 
ometry which relate to the surfaces of solids. Although articles made from 
sheet metal are hollow, being only shells, they are all considered in the pro- 
cess of pattern cutting as though they were solids. Thus the pattern for a 
cone is called the envelope of a cone, as though it were a casing stripped 
from a solid cone. 

, Ar) ... i-ii i .. ., FiQ- 2 - — Curved Lines, 

4. A t'oint is that winch has place or position without magnitude, as 

the intersection of two lines or the center of a circle ; it is therefore frequently represented to the eye by a 
small dot. 

5. A Line is measured by length merely, and may be straight or curved. 

6. A Straight Line, or, as it is sometimes called, a right line, is the shortest line that can be drawn 
between two given points. Straight lines are generally designated by 'letters or figures at their extremities 
as A B, Fig. 1. 

7. A Curve is a bending without angles. 

8. A Curved Line is one which changes its direction at every point, or one of which no portion, however 
small, is straight. It is therefore longer than a straight line connecting the same points. Curved lines are 
designated by letters or figures at their extremities and at intermediate points. (Fig. 2.) 

9. A Given Point or a Given Line expresses a point or line of fixed position or dimension. 




Definitions and Technicalities. 



A 1 - 



-B 

-B 




Fig. 3 . — Parallel Lines. 




Fig. 4. — Names of Lines by Direction. 



10. Parallel Lines are those which have no inclination to each other, being everywhere equidistant, as A 

B and A 1 B 1 in Fig. 3, which can never meet though produced to infinity. 
C D and C D 1 are also parallel lines, being arcs of circles which have a com- 
mon center. 

11. Horizontal Lines are such as are parallel to the horizon, or level. 

12. A Horizontal Line in a drawing is indicated by a line across the 
paper, as A B in Fig. 4 ; or, in other words, by a line drawn from left to right 
in front of the draftsman. 

13. Vertical Lines are such as are parallel to the position of a plumb- 
line suspended freely in a still atmosphere. 

14. A Vertical Line in a drawing is represented by a line 
drawn up and down the paper, or at right angles to a horizontal 
line, as E C in Fig. 4. 

15. Inclined Lines occupy an intermediate between hori- 
zontal and vertical lines, as G D, Fig. 4. Two lines which con- 
verge toward each other, and which, if produced, would meet or 
intersect, are said to incline to each other. 

16. Perpendicular Lines. — Lines are perpendicular to each 
other when the angles on either side of the jnoiut of junction 
are equal. Vertical and horizontal hues are always perpendicular 
to each other, but perpendicular lines are not always vertical and 

D horizontal. They may be at any inclination to the horizon, provided that 

the angles on either side of the point of intersection are equal. In Fig. 5, 
C F, D II and E G are said to be perpendicular to A B. Also in Fig. 6, 
G D and E F are pei^endicular to A B. 

17. An Angle is the opening between two straight lines which meet 
one another. An angle is commonly designated by three letters, and the 
letter designating the point in which the straight lines containing the angle 
meet, is put between the other two letters. 

18. A Bight Angle. — "When a straight 
line meets another straight line, so as to make 
the adjacent angles equal to each other, each 
angle is called a right angle, and the straight 

lines are said to be perpendicular to each other. (See C B E and C B D, Fig. 7.) 

19. An Acute Angle is an angle less than a right angle, as A B D or ABC, 
Fig. 7. 

20. An Obtuse Angle is an angle greater than a right angle, as A B E, Fig. 7. 

21. A Surface is that which has length and breadth without thickness. 

22. A Plane is a surface such that if any two of its points be joined by a 

straight line, such line will be wholly in the surface. Every surface which 
is not a plane surface, or composed of plane surfaces, is a curved surface. 

23. A Plane Figure is a portion of a plane 
terminated on all sides by lines either straight or 
curved. 

24. Pectalinear Figures. — "When surfaces 
are bounded by straight lines they are said to be 
rectalinear. (See Figs. 8, 16, 21, &c.) 

25. Polygon is the general name applied to 

all rectalinear figures, but is commonly applied to 

those having more than four sides. A regular Fig . s ._ An Equilateral 

polygon is one m which the sides are equal. Triangle. 

26. A Triangle is a flat surface bounded by three straight lines. (Figs. 8, 9, 10, 11, 13, &c.) 

27. An Equilateral Triangle is one in which the three sides are equal. (Fig. 8.) 



Fig. 5. — Perpendicular Lines. 




Fig. 6. — Perpendicular Lines. 




B 



Fig. 7. — Angles. 

C B E, C B D, right angles. A B D, A B C, acute 
angles. A B E, an obtuse angle. 




Definitions and Technicalities. 



8 



An Isosceles Triangle is one in which tvK> sides are equal. (Fig. 9.) 
A Scalene Triangle is one in which all the sides are of different lengths. 



28. 

29. A Scalene Triangle is one in which all the sides are of different lengths. (Fig. 10.) 

30. A Right-Angled Triangle is one in which one of the angles is. a right angle. (Fig. 11.) 

31. An Acute-Angled Triangle is one which has its three angles acute. (Fig. 12.) 

B B D 






Fig. n. — Right-Angled Triangles. 



Fig. g. — An Isosceles Fig. io.— ^1 Scalene 

Triangle. Triangle. 

32. An Obtuse- Angled Triangle is one which lias an obtuse angle. 

33. A Ihjpothenuse is the longest side in a right-angled triangle, or the side opposite the right angle. 
(Fig. 15.) 

34. The Apex of a triangle is its upper extremity. (Fig. 1-4.) Also called vertex. 



(F 



ig. 13.) 







Fig. 12. — An Acute- 
Angled Triangle. 



Fig. 13. — An Obtuse-Angled 
Triangle. 



Fig. 14. — Names of the Parts 
in a Triangle. 



Fig. 15. — Names of the 
Parts in a Right- Angled 
Triangle. 



35. The Base of a triangle is the line at the bottom. (Figs. 14 and 15.) 

36. The Sides of a triangle are the including lines. (Fig. 14.) 

37. The Vertex is the point in any figure opposite to and furthest from the base. The vertex of an 
angle is the point in which the sides of the angle meet. (Fig. 14.) 

38. The Altitude of a triangle is the length of a perpendicular let fall from its vertex to its base, as B D, 
Fig. 14. 

39. A Quadrilateral figure is a surface bounded by four straight lines. 







Fig. 16. — A Trapezium. Fig.iJ. — A Trapezoid. Fig. 18. — A Rhomboid. Fig. 19.— A Rhombus or Lozenge. 

There are three kinds of quadrilaterals : 

40. The Trapezium, which has no two of its sides parallel. (Fig. 16.) 

41. The Trapezoid, which has only two of its sides parallel. (Fig. 17.) 

42. The Parallelogram, which has its opposite sides parallel. 
There are four varieties of parallelograms : 

43. The Rhomb, Rhombus or Lozenge, in which the several sides are equal, and whose opposite sides are 
parallel, and in which two angles are obtuse and two acute. (Fig. 19.) 

44. The Rhomboid, which has only the opposite sides equal, the length and width being different. (Fig. 18.) 



Definitions mid Technicalities. 

45. The Rectangle, which is an equiangular parallelogram. (Fig. 20.) 

46. The Square, which is both equilateral and equiangular. (Fig. 21.) 

47. A Pentagon is a plane figure of five sides. (Fie. 22.) 

48. A Hexagon is a plane figure of six sides. (Fig. 23.) 

49. A Heptagon is a plane figure of seven sides. (Fig. 24.) 



Fig. 20. — An Equian- 
gular Parallelogram, 
called a Rectangle. 







Fig. 21. — A Square. 



Fig. 22. — A Pentagon 



Fig. 23. — A Hexagon. Fig. 24. — A Heptagon. 



50. An Octagon is a plane figure of eight sides. (Fig. 25.) 

51. A Nonagon is a plane figure of nine sides. (Fig. 26.) 

52. A Decagon is a plane figure of ten sides. (Fig. 27.) 

53. A Dodecagon is a plane figure of twelve sides. (Fig. 28.) 

54. The Peremeter of a polygon is the broken line that bounds it, as A B C D E, Fig. 22. 

55. A Diagonal is a straight line joining two opposite angles of a figure, as A B and C D, Fig. 29. 




Fig. 25.— 

56. 
(Fig. 30 

57. 

58. 
A, Fig. 

59. 
Fig. 30. 



An Octagon. 






-A Nonagon 



Fig. 28. — A Dodecagon. 



Fig. 29. — Diagonals. 



Fig. 27. — A Decagon. 

A Circle is a plane figure contained by one curved line, everywhere equidistant from its center. 

.) The term circle is also used to designate the boundary line. ^(See also Circumference.) 
The Circumference of a circle is the boundary line of the figure. (Fig. 30.) 
The Center of a circle is a point within the circumference equally distant from every point in it, as 

30. 

The Radius of a circle is a line drawn from the center to any point in the circumference, as A B, 
The plural of radius is radii. 



,»S& 






Fig. 30. — A Circle. 



Fig. 31. — A Semicircle. 



Fig. 32. — Segments. 



60. The Diameter of a circle is any straight line drawn through the center to opposite points of the cir- 
cumference, as C D, Fig. 30. The length of the diameter is equal to two radii. 

61. A Semicircle is the half of a circle, and is bounded by half of the circumference and a diameter. 
(Fig. 31.) 

62. A Segment of a circle is any part of the surface cut off by a straight line. (Fig. 32.) 



Definitions and Technicalities. 5 

63. An Arc of a circle is any part of the circumference, as A B E and F D, Fig. 33. 

64. A Sector of a circle is the space contained between two radii and the arc which they intercept, as 
A C B and D C E, Fig. 31. 

65. A Quadrant is a sector whose area is equal to one-fourth of the circle. (B A C, Fig. 35.) In a quad- 
rant the two radii are at right angles. 






Fig. 33. — Arcs and Chords. Fig. 34.— Sectors. Fig. 35. — A Quadrant. 

66. A Chord is a straight line joining the extremities of an arc, as A E and C D, Fig. 33. 

67. A Tangent to a circle or other curve is a straight line which touches it at only one point, as ED and 
A C, Fig. 36. 

68. Circles are concentric when described from the same center. (Fig. 37.) 

69. Circles are eccentric when described from different centers. (Fig. 38.) 






Fig. .jg. Tangents. Fig. 37. — Concentric Circles. Fig. 38. — Eccentric Circles. 

70. Triangular figures and those with a greater number of sides are inscribed in, or circumscribed by, 
circles when the vertices of all their angles are in the circumference. (Fig. 39.) 

71. A circle is inscribed in a straight-sided figure when it is tangent to all sides. (Fig. 40.) All regular 
polygons may be inscribed in circles, and circles may be inscribed in the polygons ; hence the facility with 
which polygons may be constructed. 

72. A Degree. — The circumference of a circle is 
considered as divided into 360 equal parts, called degrees 
(marked °). Each degree is divided into 60 minutes 
(marked '); and each minute into 60 seconds (marked "). 
Thus if the circle be large or small the number of divis- 
ions is always the same, a degree being equal to 3-J-oth 
part of the whole circumference ; the semicircle is equal 
to 180° and the quadrant to 90°. The radii drawn from 
the center of a circle to the extremities of a quadrant 
are always at right angles with each other ; a right angle 
is therefore called an angle of 90° (A E B, Fig. 41). If we bisect a right angle by a straight line, it divides the 
arc of the quadrant also into two equal parts, each being equal to one-eighth of the whole circumference, or 45", 
(A E F and FEB, Fig. 41) ; if the right angle were divided into three equal parts by straight lines, it would 
divide the arc into three equal parts, each containing 30° (A E G, G E H, H E B, Fig. 41). Thus the degrees 





Fig. 39. — An Inscribed 
Triangle. 



Fig. 40. — An Inscribed 
Circle. 



q Definitions and Technicalities. 

of the circle are used to measure angles, and when we speak of an angle of any number of degrees, it is under- 
stood that if a circle with any length of radius be struck with one foot of the compasses in the angular point, 
the sides of the angle will intercept a portion of the circle equal to the number of degrees given. Thus the 

angle A E H, Fig. -±1, is an angle of 60°. 

73. In the measurement of angles by the circumference 
of the circle, and in the various mathematical calculations 
based thereon, use is made of certain lines, always bearing 
a fixed relationship to the radius of the circle and to each 
other, which gives rise to a number of terms, some of 
which, at least, it is desirable for the pattern cutter to un- 
derstand. 

74. The Complement of an arc or of an angle is the 
difference between that arc or angle and a quadrant. In 
Fig. 42, A D B is the complement of B D C, and vice 
versa. 

75. The Supplement of an arc or of an angle is the dif- 
ference between that arc or angle and a semicircle. In 
Fig. 43, B D C is the supplement of A D B, and vice versa. 

76. A Tangent has already been defined as a straight 
line drawn without a circle, touching it at only one point. 
(See Fig. 36.) The Tangent of an Angle, or of an arc, is a 
line which touches the arc at one extremity. In Fig. 44, C B 
is the tangent of the arc E C, or of the angle E A C. 
Every tangent is perpendicular to a radius at the point it 
touches. Thus, B C is perpendicular to A C. 

77. A Secant is a straight line drawn from the center of a circle, cutting its circumference and prolonged 
to meet a tangent. (A B, Fig. 44.) 

78. The Co-Tangent of an arc is the tangent of the complement, (* <j, lug. 45.) 

79. The Sine of an arc is a straight line drawn from one extremity perpendicular to a radius drawn to the 
other extremity of the arc. (II B, Fig. 45.) 

80. The Co-Sine of an arc is the sine of the complement of that arc. 

(H K, Fig. 45.) . 

81. The Versed Sine of an arc is that part of the radius intercepted be- 
tween the sine and the circumference. (A B, Fig. 45.) 

82. The Co-Secant of an arc or angle is the secant of the complement of 
that arc or angle, as F C, Fig. 45. 




Fig. 41.— Tlie Circle Divided into Degrees, for Measuring 
Angles. 







Fig. 42. — Complement. 



Fig. 43. — Supplement. 



Fig. 44. — Secant and 
Tangent. 



Fig. 45. — Names of Lines used in 
Mathematical Calculations. 



83. In Fig. 45 are shown all the various lines and divisions appertaining to a given angle. A C II repre- 
sents the angle ; H C G is the complement of that angle, and HCD is the supplement. The names of the 
several parts are given in the diagram, and also have been defined and described above. 

84. An Ellipse is an oval-shaped curve (Fig. 46), from any point in which, if straight lines be drawn to two 
fixed points within the curve, their sum will be always the same. These two points are called .foci (F and H). 
The line A B, passing through the foci, is called the transverse axis. The line E G, perpendicular to the cen- 



Definitions and Technicalities. 



ter of the transverse axis, and extending from one side of the figure to the other, is called the conjugate axis. 
There are various other definitions of the ellipse hesides the one given here, dependent upon the means em- 
ployed for projecting it, which -will he fully explained at the proper place among the problems. 






D 
Fig. 46. — An Ellipse. Fig. 47. — .4 Parabola. Fig. 48. — A Hyperbola. 

85. A Parabola (A B, Fig. 47) is a curve in which any point is equally distant from a certain fixed point 
and a straight line. The fixed point (F) is called the focus, and the straight line (C D) the directrix. In this 
figure any point, as N or M, is equally distant from F and the same point in C D, as H or K. 

86. A Hyperbola (A B, Fig. 48) is a curve from any point in which, if two straight lines be drawn to two 
fixed points, their difference shall always be the same. Thus, the difference between E G and G L is H L, and 
the difference between E F and F L is B L. HL and B L are equal. The two fixed points, E and L, are 
called foci. 






Fig. 50. — A Triangular 
Prism. 



Fig. 51. — A Quadrangular 
Prism. 



Fig. 49. — Evolute and Involute. 

87. Aii .Evolute is a circle or other curve from which another curve, called the involute or evolutent, is 
described by the aid of a thread gradually unwound from it. (Fig. 49.) 

88. An Involute or Evolutent is a curve traced by the end of a string wound upon another curve or unwound 
from it. (Fig. 49.) 

89. A Solid has length, breadth and thickness. 

90. A Prism, is a solid of which the ends are equal, similar and parallel straight-sided figures, and of which 
the other sides are parallelograms. 

91. A Triangular Prism is one whose bases or ends are triangles. (Fig. 50.) 

92. A Quadrangular Prism is one whose bases or ends are quadrilaterals. (Fig. 51.) 







Fig. 54.-4 Cube. 



Fig. 55. — A Cylinder. Fig. 56. — A Cone. 



Fig. 52. — A Pentagonal Prism. Fig. 53. — A Hexagonal Prism. 

93. A Pentagonal Prism is one whose bases or ends are pentagons. (Fig. 52.) 

94. A Hexagonal Prism is one whose bases or ends are hexagons. (Fig. 53.) 

95. A Cube is a prism of which all the sides are square. (Fig. 54.) 

96. A Cylinder is a round solid of uniform thicknesss, of which the ends are equal and parallel circles. 
(Fig. 55.) 



3 



Definitions and Technicalities. 



97. A Cone is a round solid with a circle for its base, aud tapering uniformly to a point at the top. (Fig. 56.) 

98. A Right Cone is one in which the perpendicular let fall from the vertex upon the base passes through 
the center of the base. This perpendicular is then called the axis of the cone. (Fig. 57.) 

99. An Oblique Cone or Scalene Cone is one of which the axis is inclined to the plane of its base, and of 
which the sides are unequal. (Fig. 58.) 

100. A Truncated Cone is one whose vertex is cut off by a plane parallel to its base. (Fig. 59.) This 
figure is also called a frustum of a cone. (See Figs. 73 and 74 and accompanying definitions.) 




Base 







Fig. 57- — A Eight 
Cone. 



Fig. 58. — An Oblique or 
Scalene Cone. 



Fig. 59. — A Truncated 
Cone. 



Fig. 60. — .4 Sphere or 
Globe. 



Fig. 61. — A Trian- 
gular Pyramid. 

101. A Sphere or Globe is a solid bounded by a uniformly curved surface, any point of which is equally 
distant from the center, a point within the sphere. (Fig. GO.) 

102. A Pyramid is a solid having a straight-sided base and triangular sides terminating in one point or 
vertex. Pyramids are distinguished as triangular, quadrangular, pentagonal, hexagonal, etc., according as the 
base has three sides, four sides, five sides, six sides, etc. (Figs. 61, 62 and 63.) 



Vertex 






i<Ygr. 63. — An Octagonal 
Pyramid. 



'Base 
Fig. 64. — A Right 
Pyramid. 





Fig. 65. — Altitude 
of a Cone. 



Fig. 66. — Altitude 
of a Pyramid. 



Fig. 62. — A Quadran- 
gidar Pyramid. 

103. A Eight Pyramid is one whose base is a regular polygon, and in which the perpendicular let fall 
from the vertex upon the base passes through the center of the base. This perpendicular is then called the 
axis of the pyramid. (Fig. 61.) 

104. The Altitude of a pyramid or cone is the length of the perpendicular let fall from the vertex on the 
plane of the base. The altitude of a prism or cylinder is the distance between its two bases or ends, and is 
measured by a line drawn from a point in one base perpendicular to the plane of the other. (Figs. 65, 66, 67 
and 68.) 



\ 






Fig. 67. — Altitude 
of a Prism. 



Fig. 6S. — Altitude 
of a Cylinder. 



Fig. 69. — A Tetrahedron. 



Fig. 70. — An Octahedron. 



105. Besides the solids already described, there are others to which designating names have been applied. 

106. A Tetrahedron is a solid bounded by four equilateral triangles. (Fig. 69.) 

107. A Hexahedron is a solid bounded by six squares. The common name for this solid is cube. (See 
definition under cube, Fig. 54.) 

108. The Octahedron is a solid bounded by eight equilateral triangles. (Fig. 70.) 

109. The Dodecahedron is a solid bounded by twelve pentagons. 



Definitions and Technicalities. 



9 



110. The Icosahedron is a solid bounded by twenty equilateral triangles. 

111. An Axis is a straight line, real or imaginary, passing through a body on which it revolves, or may be 
supposed to revolve. The axis of a circle is any straight line passing through the center. The axis of a cylin- 
der is the straight line joining the centers of the two ends. (Figs. 57 and 61.) 

112. An Envelope of a solid is that which envelopes, encases or surrounds it, as the envelope of a cone. 

113. Solids are said to 'penetrate each other when they are so fitted together as to appear to pass through 
each other. Hence we have the term penetration of solids. (Fig. 71.) 







Fig. 72 



— Intersection of 
Solids. 



Fig. 73. — Frustum of a 
Scalene Cone. 



Fig. 74-— Frustum of a 
Pyramid. 



Fig. 71. — Penetration 
of Solids. 

111. Intersection of Solids is a term meaning substantially the same as penetration of solids, and is used 
to describe the condition of solids which are so joined and fitted to each other as to appear to pass through 
each other. (Fig. 72.) 

115. When a solid, as, for example, a cone, is cut through transversely by a plane parallel or inclined to the 
base, the part next the base is called a frustum of the solid. Hence we have the terms frustum cf a cone, 
frustum of a pyramid, etc. (Figs. 59, 73 and 74.) 








FiO- 75 — Conical Vhgula. 



Fig. 76. — Cylindrical Ungula. 



Fig. 77. — Cone cut by a Plane obliquely 
through its opposite sides. 



116. When a section of a solid of revolution, as, for example, a cylinder or a cone, is cut off by a plane 
oblique to the base, it is called an ungula. (Figs. 75" and 76.) 

117. A Conic Section is a curved line formed by the intersection of a cone and a plane. 

118. When a cone is cut by a plane obliquely through its opposite sides, the resulting figure is an ellipse. 
(Fig. 77.) 

D E 






/ 



Fig. 78. 



—Cone cut by a Plane parallel 
to one of its sides. 



Fig. So. — A Concave 
Surface. 



Fig. 81. — A Convex 
Surface. 



Fig. 79. — Cone cut by a Plane which 
makes an angle with the base greater 
than the angle formed by the side. 

119. When a cone is cut by a plane parallel to one of its sides, the resulting figure is a parabola. Thus 
in Fig. 78, the cutting plane A B is parallel to the side of the cone C D. 

120. When the cutting plane makes a greater angle with the base than the side of the cone makes, the 
resulting figure is a hyperbola. Thus in Fig. 79, the angle A B C is greater than the angle A D E. 



10 



Definitions and Technicalities. 



121. The parabola and hyperbola resemble eacli other, both being incomplete figures, with arms extending 
indefinitely. The ellipse is a complete figure, but of varying proportions, as the cutting plane is inclined more 
or less. 

122. Means of producing these several figures has been illustrated in Figs. 46, -±7 and 48. See also the 
accompanying definitions. Further remarks concerning the ellipse will be found among the problems. 

123. Concave means hollowed and curved or rounded, said of the interior of an arched surface or curved 
line in opposition to convex. (Fig. 80.) 

124. A Convex surface is one that is regularly protuberant or bulging, when viewed from without. The 
opposite of convex is concave. (Fig. SI.) 

125. Diamond is the name applied to a geometrical figure consisting of four equal straight lines and hav- 
ing two of the interior angles acute and two obtuse; a rhombus; a lozenge. (Fig. 18.) 

126. The term Cornice is ordinarily used to designate any molded projection which finishes or crowns the 

part to which it is affixed. Hence 
common usage accepts the term cor- 
nice in the sense of an entire entabla- 
ture, while by strict definition it is re- 
stricted to the upper division of the 
entablature as that word is understood 
in classical architecture. (Fig. 82.) 

127. An Entablature consists of 
three parts, the cornice, the frieze and 
the architrave, as illustrated by the ac- 
companying engraving (Fig. 82). 

128. The Frieze, the middle di- 
vision of the entablature (Fig. 82), is 
sometimes treated very plainly and 
sometimes receives considerable orna- 

Heuce we have the terms plain frieze 



Cornice < 



Frieze 



Architrave 




Fig. 82. — The Entablature and its Parts. 



mentation, being subdivided into panels or enriched by scrolls, etc. 

designating a frieze devoid of ornamentation ; friezepiece or frieze-panel, designating one of the parts of 

which a frieze is constructed. 

129. The Architrave is the third or lower division of the entablature. (Fig. 82.) This term is also used 
to designate a molding running around the exterior curve of an arch. 

130. Crown Molding is the term applied to the front or projecting member of a cornice. (Fig. 82.) 

131. Plamceer is the term indicating the ceiling or under side of the cornice. (Fig. 82.) 

132. Soffit is the term applied to the under side of a projecting molding. 







Fig. 83. — A Cornice Bracket. 



Fig. 84. — A Cornice Moditlion. 



Fig. 85. — A Lintel Cornice. 



133. A Drip is a downward projecting member in a cornice or in a molding, used to throw the water off 
from the other parts. (Fig. 82.) 

134. A Bracket (Fig. 83), as used in a sheet-metal cornice, is simply an ornament of the cornice. Brackets 
were originally used as supports of the parts coming above them. Modern architecture retains the form, but 
with changes in construction has kept nothing of the original use. (Fig. 82.) 

135. ModilUons are also cornice ornaments, and differ from brackets only in general shape. (Fig. 84.) 
While a bracket has more depth than projection, modillions have more projection than depth. 



Definitions and Technicalities. 



11 




136. A Lintel Cornice is a cornice covering a lintel or occurring just over an opening. This term is very 
generally used to designate the cornice used above the first story of stores. (Fig. 85.) 

137. A Deck Cornice or Deck Molding is the cornice or molding used to finish the edge of a flat roof 
where it joins a steeper portion. 

138. A Si/rik is a depression in the face of a piece of work or in a plain surface. (See face of bracket, 
Fig. 82, and side of bracket, Fig. S3.) 

139. A Truss is a large terminal bracket in a cornice, 
projecting sufficiently to receive all the moldings against its 
side, thus forming a finish to the end of the cornice. (Fig. 
86.) 

140. A Stop Block is a block-shaped structure, variously 
ornamented, which is placed above an ordinary bracket in a 
cornice, and which projects far enough to receive against its 
side the various moldings occurring above the bracket, thus 
forming an end finish to a cornice. (Fig. 87.) 

141. A Head Block is a structure in general shape and 
appearance similar to a stop block, but which, unlike the latter, 
is placed outside of«the various moldings, and whose sides 
finish against their face, forming an ornament to the crown 
molding. 

142. A Corbel is a modified form of bracket. It is used 
to terminate the lower parts of window caps, and also forms 
the support for the lower end of arches, etc., in gothic forms. 

143. A Molding is an assemblage of forms projecting be- 
yond the wall, column, etc., to which it is affixed. 

144. The Bed Moldings of a cornice are those moldings 
forming the lower division of the cornice, and which are made 
up of the bed course, modillion course and dentil course. (Fig. 82.) 

145. The Bed Course is the upper division of the bed moldings, the part with which the bracket heads 
and modillion heads ordinarily correspond, and against which they miter. (Fig. 82.) 

146. The Modillion Course of a cornice embraces those moldings which are immediately back of and below 
the modillions. It is subdivided into the modillion molding and the modillion band. (Fig. 82.) 

147. The Dentil Course of a cornice embraces those moldings 
to which the dentils are attached as ornaments, and consists of the 
dentil molding and dentil band. (Fig. 82.) 

148. Foot Molding is the common term used to designate the 
lower molding in a cornice. It is frequently in this connection used 
in the sense of architrave. (Fig. 82.) 

149. Curved Moldings are those moldings whose plan or eleva- 

Fig. 87.— A Cornice Stop-Block. t ; Qn j g a curve> 

150. A Bracket Molding, also called bracket head, is the molding around the upper part of a bracket, and 
which generally members with the bed molding, against which it finishes. (Fig. 82.) 

151. A Horizontal Molding is one whose course is in a horizontal direction. 

152. A Vertical Molding is one whose course is vertical, or at right angles 
to the horizon. 

153. An Inclined Molding is one whose course is intermediate between ver- 
tical and horizontal. 

154. A Cable Molding is an inclined molding which is used in the finish of 
a gable. 

155. A Ridge Molding is a molding used to cap or finish a ridge. It is also called a ridge capping, or sim 
ply ridging. 

156. A Hip Molding is a molding used to protect and finish the hips or angles of a roof. It is very ire- 
queutly included in the more general term ridging. 



Fig. 86. — A Cornice Truss. 






Fig. 88.— Stays or Profiles. 



12 



Definitions and Technicalities. 



A 



157. The Face of a molding is its outer surface when placed in the position it is intended to occupy. 

158. The Stay of a molding is its shape or profile cut in sheet metal. (Fig. 88.) 

159. Rake Holdings are those which are inclined, as in a gable or pediment ; but since to miter a rake 
molding or an inclined molding, under certain conditions, necessitates a change or modification of profile in one 
or the other of the moldings, to rake has come to mean to make such change of profile. 

160. A Raked Molding, therefore, is a term describing a molding of which the profile 
is a modification of some other profile. 

161. A Raked Profile or Raked Stay describes the profile or stay which has been de- 
rived from another profile or stay, by certain established rules, in a process like that of 
mitering a horizontal and inclined molding together. 

162. The Normal Profile or Normal Stay is the original profile or stay from which the 
raked profile or stay has been derived. 

163. The term Miter designates a joint in a molding, or between two pieces not mold- 
ings, at any angle. 

161. A Butt Miter is the term applied to the cut made upon the end of a molding to 
fit it against another molding or against a surface. 

165. A Gable Miter is the name applied to the miter either at the peak or at the foot 
of a gable molding. 

166. A Rake Miter is a miter between two moldings, one of which has undergone a 
modification of profile to admit of the joint being made. 

167. Square Miter is the common term for a joint at right angles, or at 90°. 

168. An Octagon Miter is a miter joint between two sides of a regular octagon, or be- 
tween any two pieces at an angle of 135°. 

ig. 9.— ^ nnac e. jg<^ ^ n j ns i r j e 2£iter indicates a joint at an interior or re-entrant angle, 

170. An Outside Miter is a joint at an exterior angle. 

171. A Miter Piece is one of the pieces next the proper cut made ivpon it, between 
which a miter joint is to be made. 

172. A Complete Miter is the structure formed by the union of two pieces of mold- 
ing by means of a miter joint. 

173. A Fillet is a little square member, and is of frequent occurrence in moldings. 

174. A Flange is a projecting edge by which a piece is strengthened or fastened to 
anything. 

175. A Pinnacle is a slender turret or part of a building elevated above the main build- 
ing. (Fig. 89.) Fig. 90.— A Pilaster. 

176. A Pilaster is a square column, usually set within a wall, and projecting part of its diameter. (Fig. 90.) 

177. A Pediment is a triangular ornamental facing of a portico, or a similar decoration over doors, win- 
dows, etc. The name is also applied to arched ornaments of a like kind. (Figs. 91 and 92.) 



^ , ""T'' L r=^T^™ f 






i i 


i 




1 ' 


i 




• 1 






, 1 








, 1 


hj 


1 






1 




I 






i 




=\.i 





Fig. 91.— An Angular Pediment. Fig. 92. — A Segmental Pediment. 

178. A Broken Pediment is one, either in the form of a gable or arch, which is cut away in its central 
portion for the purpose of ornamentation. (Fig. 93.) 

179. An Elevation is a geometrical projection of a building or 
other object on a plane perpendicular to the horizon. (Fig. 94.) 

180. A Plan is the representation of the parts as they would 
appear if cut upon a horizontal line. (Fig. 95.) 

181. A Section is a view of the object as it would appear if cut 
in two at a given vertical or horizontal plane. (Fig. 96.) In the 

one case the resulting figure is called a vertical section, and in the other a horizontcd section. Oblique sections 
are representations of objects cut at various angles. 




Fig. 93. — A Broken Pediment. 



Definitions and Technicalities. 



13 



182. A Perspective is a representation of a building or other object upon a plane surface, presenting the 
same appearance that the object itself would present if viewed from a particular point. (Fig. 97.) 
1S3. A Draft is a figure described on paper. 

184. A Drawing is a rej)resentation on a plane surface, by means of lines, or by means of lines and shades 
of the appearance or figure of objects. 

185. A Detail Drawing is a drawing commonly full size, for the use of mechanics in constructing work. 

186. A Working Drawing is the same as a detail drawing. 

187. A Scale Drawing is one made to some scale less than full size. 




■ 

T> 



Fig. 94. — Elevation of House. 



Fig. 95-— Pfa» of House. 



188. Incised Work is a style of ornamentation consisting of fine members and irregular lines, sunken or 
Cut into a plane surface. 

189. A Hip is the external angle formed by the meeting of two sloping sides or skirts of a roof which 
have their wall plates running in different directions. 

190. A Gable is the vertical triangular end of a house or other building, from the cornice or eaves to the top. 

191. A Problem is a question proposed for solution. This term also describes anything which is required 
to be done, as to bisect a line. 

192. A Proposition is that which is offered for consideration or adoption — a statement in terms, either of 
a truth to be demonstrated or of an operation to be performed. 

193. An Hypothesis is something not proved, but assumed for the sake of argument ; a supposition. 





Fig. 96. — Section of House on Line A B. 



Fig. 97.— Perspective View of House. 



194. A Demonstration is a course of reasoning showing that a certain result is the necessary consequence 
of assumed premises. 

195. A Premise is something previously stated or assumed as ground for further argument. 

196. A Basis of anything is its groundwork or first princij^le. 

197. A Conclusion is the final decision or determination. 

19S. To Develop a pattern is to define its shape and boundaries by a series of progressive steps. 
199. The Development of a surface is the process of changing a given surface into another form of equiv- 
alent area or value. 



14 Definitions and Technicalities. 

200. To Project a figure is to construct, by means of lines, etc., on paper, a representation of the figure as 
it would appear from a given point of sight. 

201. Ratio is the relation which one quantity or magnitude has to another of the same kind, as expressed 
by the quotient of the second divided by the first. Thus the ratio of 4 to 8 is expressed by f or 2. 

202. The Area of a figure is its superficial contents, or the surface contained within its boundary lines. 

203. To Raise, means to form or to shape by hammering or stamping. 

204. Bisect, signifies to divide into two equal parts, or, in other words, to cut in the middle. 

205. Prolong, means to continue in the same direction ; to draw still further. 

206. Indefinitely, signifies without a limit. To prolong a line indefinitely means to draw it without regard 
to a limit of length. 



DRAWING TOOLS AND MATERIALS. 



207. In the following description of the appliances, tools and materials likely to be of service to the pat- 
tern cutter, we purposely omit all mention of some special tools, although in general use, because they are not 
likely to be of service in the class of work in which the reader is supposed to be most interested. "We limit 
our description, therefore, to articles of general use to the pattern cutter, rather than extend them into a general 
treatise upon drawing tools and materials. We shall only treat exhaustively such topics as are of special value 
to the pattern cutter. All others will be discussed only so far as they are likely to interest the special class for 
which the book is prepared. Those who are interested in drawing tools and materials upon a broader basis than 
here presented, are respectfully referred to special treatises on drawing and to the catalogues of manufacturers 
and dealers in drawing materials and drawing instruments. 

208. Drafting Tables. — A drafting table suitable for a jobbing shop should be about five feet in length 
and three to four feet in width. It is better 
to have a table somewhat too large, than to 
have one so small that it is frequently inad- 
equate for work that comes in. In Light 
the table should be such that the draftsman, 
as he stands up, may not be compelled to 
stoop to his work. While for some reasons 
it is desirable that the table should be fixed 
upon a strong frame and legs, for conven- 
ience such tables are generally made porta- 
ble. Two horses are used for supports and 
a movable drawing board for the top. A 
shallow drawer is hung by cleats fastened 
to the under side, and is arranged for pulling 
either way. Sometimes horizontal pieces Fig - &■— Drafting Table. 

are fastened to the legs of the horses, and a shelf or shelves are formed by laying boards upon them. In Fig. 
98 we show such a table as we have just described. When properly made, using heavy rather than light mate- 
rial, such a table is quite solid and substantial, yet when not in use can be packed away into a very small space. 

209. For cornice makers' use, a table similar in all respects to the one we have described and illustrated 
(Fig. 98), except in size, is well adapted. Its dimensions, considering the extremes of work that are likely to 
arise, should be twelve to fourteen feet in length by about five feet in breadth. Three horses are necessary, and 
two drawers may be suspended. With very large work, one draftsman or pattern cutter will require the whole 
table, but for ordinary work, such as window caps, cornices, etc., two men can work at it without interfering 
with or incommoding each other. 

210. Various woods may be used for drawing tables, but white pine is the cheapest and best for the pur- 
pose. Inch and one-half to two-inch stuff will be found economical, as it allows for frequent redressing — made 
necessary by pricking in the process of pattern cutting. Narrow stuff, tongued and grooved together or joined 
by glue, is preferable to wide plank, as being less liable to warp. Rods run through the table edgeways, as 
shown in Fig. 98, are desirable for drawing the parts together and holding in one compact piece. The nut and 
washer are sunken into the edge of the table, a socket wrench being used to operate them. 




16 



Drawing Tools and Materials. 




1 — £>■ 



Fig. 99. — Drawing Board, with Ledges. 



211. Each drafting table should be an accurate rectangle. Every corner should be a right angle, and the 
opposite sides should be parallel. The edges should be exactly straight throughout their length. Methods of 
testing drafting tables and drafting boards, with reference to these points, will be found on the ojmosite page. 
The usual way of adjusting a table or board to make it accurate, is to plane off its edges as required. But 
this is a task less simple and easy than it appears. It requires the nicest skill and accuracy to render it at 
all satisfactory. When it is remembered that no matter how well seasoned the lumber emplo} r ed may be, 
the table will be affected by even slight changes in the atmosphere, it is apparent that dressing off the edges with 
a plane, under certain circumstances, might be constantly required. Hence, in some of the best shops, an 
adjustable metal strip is fastened to the edge of the table in such a manner that, by simply turning a few screws, 

any variation in the table may be compensated. This metallic edge is variously 
constructed. One of the simplest forms is described as follows : The edge 
of the table on all sides is cut away so as to allow a bar of steel, say one-eighth 
or one-sixteenth of an inch thick and about an inch wide, to lie in the cutting, 
so that its surface is even with the face of the table, with one edge pro- 
jecting somewhat beyond the edge of the table. Slotted holes are made in 
the table, through which bolts with countersunk heads are passed for holding the steel strips. A washer and nut 
are used on the under side of the table. The adjustment required is very slight, so that this arrangement works 
very well, although other and more accurate methods, and more expensive also, are in use. Any plan sim- 
ilar to this will be found very useful. Iron instead of steel, if planed accurately, can be made to answer a 
good purpose. The edge of the metal projecting slightly, as we have described, is well adapted for receiving the 
the head of the T square, rendering the use of that instrument more satisfactory than when it is used against 
the plane edge of the table, even if equally accurate. 

212. Drawing Boards. — For a pattern cutter's use, the princi- 
pal difference between a drafting table and a drawing board is in 
the size. The same general requirements in point of accuracy, etc., 
are necessary in each. We have indicated convenient sizes of 
tables for various uses in our remarks under drafting tables, but to 
point out sizes of boards for different purposes is not so easy a 
matter, their application being far more extended and their use 
more general. A drawing board may be made of any required 
size, from the smallest for which such an article is adapted, up to the extreme limit consistent with convenience 
in handling. In the larger sizes the general features of construction noted under drafting tables are entirely 
applicable, save that thinner material should be used in-order to reduce the weight. In small sizes there is the 
choice between several different modes of construction. We shall describe but two or three of them, remark- 
ing that boards of almost any required construction can be purchased ordinarily of dealers in drawing tools 
and materials at lower prices than they can be made. However, it is very convenient, in many cases, to have 
boards made to order, and therefore detailed descriptions of good constructions are desirable. Any carpenter or 
cabinet maker should be able to make such boards as we present. 

213. In Fig. 99 is shown a very common form of drawing board, being a pine-wood top with hard-wood ledges. 

The ledges are put on by means of a tapering dovetail, and are 
so arranged that while allowing entire freedom for seasoning, 
so that there is no danger of cracking the board, they may be 
driven tight as required. Where it is desirable to use screws 
in the ledges, they are passed through slotted holes furnished 
with a metallic bushing. 

214. In Fig. 100 is shown a still simpler form of board, which 
is adapted only for the smallest sizes. The edges are clamped 
by hard-wood strips, as shown in the engraving. By using 
Fig. 101.— Drawing Board, with Grooved Back and Ledges strips of wood thicker than the board, keej>ing their upper sur- 
put on with slotted Holes. faces flugh ^^ the sur f ace f t h e \) 0a , r ^ this style is sometimes 

constructed so as to have the advantage of ledges on the under side equivalent to those shown in Fig. 99. 

215. Fig. 101 shows a construction of a board which, while being somewhat more expensive than others, is 
undoubtedly much better. It is made of pine wood, glued up to the required width. A pair of hard-wood 




Fig. 100. — Drawing Board, with Clamped Edges. 




Drawing Tools and Materials. 



17 




Fig. 102. — Testing the Opposite Sides of a 
Drawing Board. 



ledges are screwed to the back, the screws passing through the ledges in oblong slots with brass bushings which 
fit closely under the heads and yet allow the screws to move freely when drawn by the shrinkage of the board. 
To give the lodges power to resist the tendency of the surface to warp, a series of grooves are sunk in half the 
thickness of the board over the entire back. These grooves take the transverse strength out of the wood to 
allow it to be controlled by the ledges, leaving at the same time the longitudinal strength of the wood nearly 
unimpaired. To make the two working edges perfectly smooth, allowing an easy movement with the T-square, 
a strip of hard wood is let into the end of the board. This strip is afterward sawn apart at about every inch 
to admit of contraction. In the construction of such boards, additional advantage is obtained by putting the 
heart side of each piece of wood to the surface. 

216. Boards of the several kinds described above use the paper fastened to them, either by means of 
tacks or by gluing. Boards are sometimes made with a hard-wood frame, the board proper fitting into it as 
a panel. It is fastened into the frame by means of buttons. The paper is spread over the board, the frame 
passed down around it, carrying the edge of the paper with it, and when 
in proper position the buttons are turned into their places. Such boards 
are not well adapted to practical use. It is more difficult to stretch the 
paper by means of them than it would seem by the description. A con- 
siderable waste of paper is involved, and the necessary play of the parts, 
to allow room for the paper between them, sometimes leads to inaccu- 
racies in drawings. Stretching paper by gluing or by the use of thumb 
tacks is found far more satisfactory. For such drafting as the pattern 
cutter is called upon to perform, thumb tacks are used almost exclusively. For architectural drawings and 
for drawings of machinery, usually made on white paper, gluing is preferable. 

217. Testing Drawing Boards and Tables. — The great desideratum in a drawing table or board is its 
accuracy. It should be an accurate rectangle, in order to- facilitate work that is to be done upon it. If each 
angle is a right angle — if its opposite sides are exactly parallel — the T-square may be used at will from any por- 
tion of it with satisfactory results. If the board is accurate the drawing will be accurate. If the board is not 
accurate the drawing can only be made accurate at the cost of extra trouble and care. While it is easy to get 
a board approximately correct by ordinary means, one or two simple tests, which we shall describe, serve to 
point out inaccuracies for correction which by ordinary means would pass unnoticed. "We assume at the outset 
that we have a T-square and an ordinary two-foot steel square that are exactly correct. 

• 218. Having made the opposite sides and ends of the board as nearly accurate as possible, place the head 
of the T-square against one side, as shown in Fig. 102, and with a hard pencil sharpened to a chisel edge, or 
with the blade of a knife, scribe a fine line across the board. Then carrying the T-square to the opposite side 
of the board, as shown by the dotted lines, bring the edge of the blade to within a short distance of the line 

just described and draw another. If the two lines are found upon meas- 
urement to be exactly parallel, it is satisfactory proof that the opposite 
edges of the board are parallel at the points tested. Instead of drawing 
the lines a short distance apart, they may be drawn at the same point ; then 
instead of measuring, it will be necessary simply to see that they exactly 
coincide throughout their length. Kepeat this operation at frequent inter- 
vals along the edges of the board, both at the sides and ends. Remove 
any small inaccuracies by means of a file, or fine sand paper folded over 
a block of wood. Careful work in this manner will produce very satisfactory results. 

219. A means of testing a board with reference to the accuracy of the corners, is shown in Fig. 103. A 
carpenter's try-square or an ordinary steel square used upon the corners, does not ordinarily reach far enough in 
either direction to satisfactorily determine that the adjacent end and side are perpendicular to each other ; hence 
it is desirable to obtain some kind of a test with reference to this point from the central portions of the edges. 
With the head of the T-square placed against one side of the board draw a fine line, as indicated by the dots in 
the engraving, and from one end draw a second line in the same manner. If the side and end are at right 
angles, the two lines will correspond with the arms of a square when placed as shown in the engraving. Repeat 
this operation for each of the corners. 

220. The two methods above described for testing drawing boards, especially when used together, cannot 
fail to enable any one to obtain a board as nearly accurate as it is possible to make things accurate by ordinary 




Fig. 103. — Testing the Corner of a 
Drawing Board. 



o 







18 Drawing Tools and Materials. 

mechanical means and of the materials used. Modifications of the methods here given, and based upon the 
same principles, will suggest themselves to any one who will give the matter careful thought. 

221. Straight-Edges.— -In connection with every set of drawing instruments there should be one or more 
straight-edges. If nothing but pencil or pen lines are to be made upon paper, hard wood or hard rubber as a 

material will answer very well ; but if lines are to be drawn 
upon metal, steel is the only satisfactory material. The 
length of the straight-edge must be determined by the 
ig. 104.— rmg .- gt. work to be done, but a safe rule is to have it somewhat 

near the length of the table or board. Of course this is out of the question in cornice work, where tables are 
frequently upward of twelve feet in length. In such cases the size of the material to be cut determines this 
matter. If iron 96 inches long is used, the straight-edge, for convenience, should be not less than Si feet. If 
shorter iron is regularly used, a shorter straight-edge will answer. In cornice work, two and even three different 
lengths are found advantageous. The longest we have just described ; a second might be about four feet in length 
and made proportionately lighter, while the smallest might be two feet and also still lighter than the four-foot 
size. For the latter, however, the long arm of the common steel square serves a good purpose. 

222. For tinners' use in general jobbing shops, a three-foot straight-edge in many cases, and a four-foot one 
in a few instances, will be found quite convenient. Some mechanics desire their straight-edges graduated, the 
same as a steel sqiiare, into inches and fractions. "We see no special advantage in this ; it adds considerably 
to the cost, without rendering the tool more useful. 

223. A hole should be provided in one end of the straight-edge for hanging 
up. It should always be suspended when not in use, as in that position it is not 
liable to receive injury from any cause whatever. 

224. It is almost superfluous to add that straight-edges must be entirely ac- 
curate, for if inaccurate they would belie their name. A simple and convenient '"' ° 5 ' ixe ' ea 2 uare - 
method of testing straight-edges is to place two of them together by their edges, or a single one against the 
ede;e of a square, as shown in Fig. 107, and see if light passes between them. If no space is to be observed 
between the edges, it is satisfactory evidence that they are as straight as they can be made by ordinary appli- 
ances. In addition to having the edges straight, it is also necessary to have the two sides parallel. 

225. ~\-Squares. — "With this instrument, as with almost all drawing instruments, there is the choice of 
various qualities, sizes and kinds, and selection must be made with reference to the kind of work that is to 
be performed. Whatever quality may be chosen, the desirable features of a T-square are strict accuracy in all 
respects, a thin blade and one that will lie close to the paper when in use. For most purposes a fixed head, as 
shown in Fig. 105, is preferable. For drawings in which a great number of parallel oblique lines are required, 
and particularly where a small size T-square can be used, a swivel head, as shown in Fig. 106, is sometimes 

desirable. The objectionable feature about a swivel head is the difficulty 
of obtaining positive adjustment. When made in the ordinary manner, and 
depending upon the friction of the nut of a small bolt for holding the head 
in place, it is almost impossible to obtain a bearing that can be depended 
upon during even a simple operation. In practice it is found to be far less 
trouble to work from a straight-edge — properly placed across the board and 
weighted down or otherwise held in place — by means of a triangle or set- 
square. Greater accuracy is also assured by this plan. 

226. In point of materials, probably a T-square constructed with walnut head and maple blade is as 
likely to give good satisfaction as any. This kind is the cheapest, and is generally considered the best for 
practical purposes. A good article, but of a higher price, consists of a walnut head with a hard-wood blade, 
lined with some other kind of wood. Still another variety has a mahogany blade lined with ebony. T-squares, 
constructed with cast-iron head — open work finished by japanning — with a nickel-plated steel blade, are also 
to be had from dealers. For accuracy probably these are the best, but they are several times more expensive 
than the simple wooden material first above described. 

227. T-squares are also made with a hard rubber blade, of which Fig. 106 is an illustration. The liability 
to fracture, however, by dropping necessitates the greatest care in use ; otherwise hard rubber makes a very 
desirable article, and is the favorite material with many draftsmen. 

228. In point of size, T-squares should be selected with reference to the use to be made of them. Gener- 




Drawing Tools and Hater! ah. 



19 



ally, the blade should be a very little less in length than the width of the table or board upon which it is to be 
used. Where a large board or a table is used, it will be found to be economy to have two instruments, one 
large one and one small one, the former being used for the principal lines in laying off the work, while the 
latter is used in miter cutting and wherever the diagram can be made near enough the edge of the table to 
admit of its employment. 

229. The Steel Square. — One of the most useful tools in connection with the pattern cutter's outfit is an 
ordinary steel square. The divisions upon it concern him much less than its quality in the way of accuracy. He 
seldom requires other divisions than inches and eighths of an inch ; therefore in selection the principal point to 
be considered is that of accuracy. The finish, however, is a matter not to be overlooked. Since a nickel-plated 
square costs but a trifling advance upon the plain article, it is cheaper in the long run to have the plated tool. 



'I'l'l'HI'l'I'l 



Cprn 
I.T.l.l 



[llH'Tlili'l'I'lUil'I'I'l 

I . I ■ I ■ 1 ■ t ■ 1 ■ I ■ 1 , 1 1 1 II I. '.I. M.I 



■ I' ri ' i ' i ' i'i ' i ' i'i ' ri ' ri ' i 'i' i ' i 'i'i'i'i'U 
. i.i.i . i.i.i ..i.iiiii.i.i.i.ii.i.i.i.i.i.i.i.i.i.i. i.ii.i.i.i.i.i.i.i.i 



i ■ i ■ i ■ i ■ i ■ i ■ -i ■ i 

i.t.i.i.i.i.i.i.i.i. i. i.i.i. i.i. i.i.i. 



Fig. 107. — Testing the Exterior Angle of Fig. 108. — Testing the Interior Angle of a 

a Steel Square. Steel Square. 

230. A convenient method of testing the correctness of the outside of a square, and one which can be 
used at the time and place of purchase, is illustrated in Fig. 107. Two squares are placed against each other 
and against a straight-edge, or against the arm of a third square. If the edges exactly coincide throughout, the 
squares may be considered correct. 

231. Having procured a square which is accurate iipon the outside, the correctness of the inside of another 
square may be proven, as shown in Fig. 10S. Place one square within the other, as shown. If the edges fit 
together tightly and uniformly throughout, the square may be considered entirely satisfactory. 

232. An accurate square is especially desirable, as it affords the readiest means of testing the T-square and 
the drawing table or board, as elsewhere described. The greatest care should be given, therefore, to the selec- 
tion of a square. For all ordinary purposes the two-foot size"is most desirable. In some cases the one-foot size 
is better suited. Many pattern cutters on cornice work like to have both sizes at their command, making use of 
them interchangeably, according to the nature of the work to be done. 






B— 



Fig. no. — Hard Wood Triangle, or 
Set-Square, 30, 60 and go degrees. 



Fig. in. — Testing a Right-Angled Triangle, 
or Set-Square. 



Fig. 109. — Open Hard Rubber Triangle, 
or Set-Square, 45, 45 and go degrees. 

233. Triangles, or Set Squares. — In the selection of triangles, the draftsman has the choice in material be- 
tween pear wood ; mahogany, ebony lined ; hard rubber ; German silver ; and steel, silver or nickel plated. In 
style he has the choice between open work, of the general form shown in Fig. 109, and solid, of the general form 
of Fi°\ 110. In shape, the two general kinds which are adapted to the pattern cutter's use we have shown in 
Figs. 109 and 110, the latter being commonly described as 30, 60 and 90 degrees, and the former as 45, 45 and 
90 degrees. The special uses of each of these two articles are shown among the problems. In size, the pattern 



Drawing Tools and Materials. 



cutter requires large rather than small. If he can have two sizes of each, the smaller might measure from 4 to 
6 inches on the side, and the larger from 10 to 12 inches ; but if only a single size is to be had, one having 
dimensions intermediate to those named will be found most serviceable. 

234. The value of a triangle, for whatever purpose used, consists of its accuracy. Particularly is this to be 
said of the right angle, which is used more than either of the others. A method of testing the accuracy of the 
right ancle is shown in Fig. 111. Draw the line A B with an accurate ruler or straight-edge. 



Place the right 

ano-le of a triangle near the center of this line, and make one of the edges coincide with the line, and then 
against the other 



edge draw the lino D C. Turn the triangle on this perpendicular line, bringing it into the 
position indicated by D C A. If it is found that the sides agree with A C and C D, it is proof that the angle 
is a right angle and that the sides are straight. 

235. Besides the kinds of triangles we have described above, a fair article can be made by the mechanic 
from sheet zinc or a heavy piece of tin. Care need only be taken in cutting to obtain the greatest possible 
accuracy. For many of the purposes for which a large size 45- degree triangle would be used, the steel square 
is available ; but as the line of the hypothenuse is lacking in it, it cannot be considered a substitute. 

236. Compasses and Dividers. — The term compasses is applied to those tools, of various sizes and descrip 
tions, which hold a pencil or pen in one leg, while dividers are those tools which, while of the same general 
form as compasses, have both legs in the shape of fixed points. They derive their name from their obvious 
use, that of spacing or dividing. A special form of dividers — used exclusively for setting off spaces, as in the 
divisions of a profile line — is called spacers, as illustrated and described below. 





Fig. 112. — Compasses, with Needle Point, 
Pencil Point, Pen and Lengthening Bar. 



Fig. 113. — Plain 
Dividers. 



Fig. 114. — Hair-Spring 
Dividers. 



Fig. 115.— Steel Spring 
Spacers. 

237. A pair of compasses consists of the parts as shown in Fig. 112, being the instrument proper with 
detachable points, and extras comprising a needle point, a pencil point, a pen and a lengthening bar, all as shown 
to the left. In selection, care should be given to the workmanship ; notice whether the parts fit together neatly 
and without lost motion, and whether the joint works tightly and yet without too great friction. A good Ger- 
man silver instrument, although quite expensive at the outset, will be found the cheapest in the end. A pencil 
point of the kind shown in our engraving is to be preferred over the old style which clamps a common pencil 
to the leg. The latter is not nearly so convenient and is far less accurate. 

238. Of dividers there are two general kinds, the plain dividers, as shown in Fig. 113, and the hair-spring 
dividers, as shown in Fig. 114. The latter differ from the former simply in the fact of having a fine spring 
and a joint in one leg, the movement being controlled by the screw shown at the right. In this way, after 
the instrument has been set approximately to the distance desired, by means of the screw the adjustable leg is 
moved, as may be required, either in or out, thus making the greatest accuracy of spacing possible. Both in- 
struments are found desirable in an ordinary set of tools. The plain dividers will naturally be used for larger 
and less particular work, while the hair-spring dividers will be used in the finer parts. It frequently happens 
that two pairs of dividers, set to different spaces, are convenient to have at the same time. Then the possession 
of these two articles is especially desirable. 

239. A pair of spacers, shown in Fig. 115, is almost indispensable in a pattern cutter's outfit. He will find 
advantageous use for this tool, even though possessing both pairs of dividers described above. ■ In size it should 



Drawing Tools and Materials. 



21 



be a little less than that of the dividers. The points should be needle-like in their fineness, and should be capa- 
ble of adjustment to within a very small distance of each other. It is sometimes desirable to divide a given 
profile into spaces of an eighth of an inch. The spacers should be capable of this, as well as adapted to spaces 
of three-quarters of an inch, without being too loose. As will be seen from the engraving, this instrument is 
arranged for minute variations in adjustment. It has a marked advantage over the hair-spring dividers, in that 
the legs are controlled by the spring and screw direct ; in the latter but one leg is affected by the spring, leaving 
only the friction of the joint to keep the legs in one constant position relative to each other. 

240. Beam Compasses and Trammels. — In Fig. 116 we show a set of beam compasses, together with a 
portion of the rod or beam on which they are 
used. The latter, as will be seen by the section 
drawn to one side (A), is in the general shape 
of a T. This form has considerable strength 
and rigidity, while at the same time it is not 
clumsy or heavy. Beam compasses are pro- 
vided with extra points, for pencil and ink 
work, as shown. While the general adjust- 
ment is effected by means of the clamp against 
the wood, minute variations are made by the 
screw shifting one of the points, as shown. 
This instrument is quite delicate, and when 
in good order is very accurate. It should be 
used only for fine work on paper, and never 
for scribing on metal. 




Fig. ri6. — Beam Compasses. 



241. A coarser instrument, and one especially designed for use upon metal, is shown in Fig. 117, and is 
called a trammel. It is to be remarked in this connection that the name trammel, by common usage, is applied 
to this instrument and also to a device for drawing ellipses, which will be found described at another place. 
There are various forms of this instrument, all being the same in principle ; our engraving shows one that is 
in quite common use. A heavier stick is used with it than with the beam compasses, and no other adjust- 





Fig. 117. — Trammel. Fig. 118. — Trammel, Slioieing Method of Using Pencil. 

ment is provided than that which is afforded by clamping against the stick. In Fig. 117 a carrier at the side 
is shown, in which a pencil may be placed. Some trammels are arranged in such a manner that either of the 
points may be detached and a pencil substituted. In others it is intended that the pencil shall be placed in a 
side carrier without removing the point. In Fig. 118 we show the form of trammel just described, arranged for 
using a pencil. 



22 Drawing Tools and Materials. 

242. A trammel, by careful management, can be made to describe very accurate curves, and hence can be 
used in place of the beam compasses in many instances. For all coarse work it is to be preferred to the beam 
compasses. It is useful for all short sweeps upon sheets of metal, but for very long sweeps a strip of sheet iron 
or a piece of wire will be found of more practical service than even this tool. 

243. The length of rods for both beam compasses and trammels, up to certain limits, is determined by the 
nature of the work to be done. The extreme length is determined by the strength and rigidity of the rod 
itself. It is usually convenient to have two rods for each instrument, one about 3i or 4 feet in length and 
the other considerably longer— as long as the strength of material will admit. In the case of the trammel, by 
means of a simple clamping device, or, in lieu of better, by use of common wrapping twine, the rods may be 
spliced when unusual length is required ; but, as remarked before, a strip of sheet iron or a piece of fine wire 
forms a better radius, under such circumstances, than the rod. 

244. The Protractor is an instrument for laying down and measuring angles upon paper. The instrument, 

when by itself, consists of a semicircle 
of tnin metal or horn, as represented 
in Fig. 119, the circumference of 
which is divided into ISO equal parts 
or degrees. The principles upon which 
the protractor is constructed and used 
are clearly explained in the chapter of 
definitions, under the head "Degree." 
The methods of employing it in the 
construction of geometrical figures are 
shown in the proper place among the 
problems. For purposes of accuracy, 
a large protractor is to be preferred 
to a small size, because in the former 

Fig. n 9 -A Semicircular Protractor. fractions of a degree are indicated. 

245. While a number of geometrical problems are conveniently solved by the use of this instrument, it is 
not one that is specially adapted to the pattern cutter's use. All the problems which are solved by it are 
capable of other accurate and expeditious methods, which, in most cases, are preferable. It is one of the in- 
struments, however, included in almost every case of instruments sold, and the student will find it advantageous 
to become thoroughly familiar with it, whether in practice he employs it or not. 

246. Besides the semicircular form of the protractor shown in Fig. 119, corresponding lines and divisions 
to those upon it are sometimes put upon some of the varieties of scales in use, allusion to which will be found 
in our remarks upon scales. 

247. Scales.— Many of the drawings from which the pattern cutter works— that is, from which he gets 
dimensions, &e. — are what are called scale drawings, being some specified fraction of the full size of the object 
represented. Architects' elevations and floor plans are very generally made either I or \ inch to the foot, or, 
in other words, 1-96 or 1-48 full size. Scale details are also employed quite extensively by architects, scales in 
very common use for the purpose being \\ 

inches to the foot and 3 inches to the foot, i,,,,,,,,,,,,,,, i ^ » 3 

or, in other words, -J- and \ full size respec- l ihhln 




tively. It is essential that the pattern cut- Fig. 120.— Plain Scale. — 1 inch to the foot. 

ter should be familiar with the various 

scales in common use, that he may be able to work from any of them on demand. Several of the scales are 
easily read by means of the common rule, as, for example, 3 inches to the foot, in which each quarter inch on 
the rule becomes one inch of the scale ; also, 1£ inches to the foot, in which each eighth of an inch on the rule 
becomes an inch of the scale ; and, likewise, f inch to the foot, in which each sixteenth of an inch on the rule 
becomes an inch of the scale. However, other scales besides these are occasionally required, which are not 
easily read by the common rule, and sometimes special scales are used which are not shown on the instru- 
ments especially calculated for the purpose. Accordingly, it is sometimes necessary for the pattern cutter to 
construct his own scale. 

248. The method of constructing a scale of 1 inch to the foot is illustrated in Fig. 120, in which the divi- 



Drawing Tools and Materials. 



23 



sions are made by feet, inches arid half inches. In constructing such scales, it is usual to set off one division at 
the left, as shown, for division into inches and fractions of an inch. In using the scale, measurements are made 
to commence with the second division. When the number of feet has been found in this way, the instrument 
is shifted from left to right until the nearest division of feet comes opposite the end of the space measured ■ 
the feet are read by the number thus found, while a glance at the other end of the rule shows how many inches 
constitute the fraction of the foot. 

249. Besides scales of the kind just described, which are termed plain divided scales, there are in common use 
what are known as diagonal scales, an illustration of one of which we show in Fig. 121. The scale repre- 
sented is 1^ inches to the foot. 
The left-hand unit of division 
has been divided by means of the 
vertical lines into 12 equal parts, 
representing inches. In width 
the scale is made to equal 8 of 
these parts, and the intermediate 
parallel lines arc drawn. Next 
the diagonal lines are drawn. By 
a moment's inspection it will be 
seen that, by means of these diagonal lines, one-eighth of an inch and multiples thereof are shown on the several 
horizontal lines. If we have a distance equal to the space from A to B, as marked on the scale, we read it (first 
at the right for feet) 2 feet, (then in the left for inches by means of the vertical lines figured both at top and 
bottom) 6 inches (and last by means of the diagonal line, figured at the end of the scale, for fractions) and three- 
eighths. The top and bottom lines of the scale measure feet and inches only. The other lines measure feet, 
inches and fractions of an inch, each horizontal line having its own peculiar fraction, as shown. Such scales are 
frequently quite useful, and, as the reader will see, may be constructed by any one to any unit of measurement, 
and divided by diagonal lines into any desired fractions. 













, 


















1 




1 








' 




1 


1 






1 










B 










1 








1 


1 






' 


/ 








1 






















1 
















1 
















' 




















I 












/ 





















' 















|2 11 10 9 



Fig. 121. — Diagonal Scale. — 1% inches to the foot. 





\ \ \ \ \ \ \ \ I 1 / / / / ,-- - / ~s / 



Fig. viz. — Triangular Boxwood Scale. Fig. 123. — Flat Boxwood Scale. 

250. A scale in common use. and known as the triangular scale, is shown in Fig. 122. The shape of this 
scale, which is indicated by the name, and which is also shown in the cut, presents three sides for division. By 
dividing these through the center lengthways by a groove, as shown, six spaces for divisions are obtained, and 
by running the scales in pairs — that is, taking two scales, one of which is twice the size of the other — and com- 
mencing with the unit at opposite ends, the number of scales which may be put upon one of these instruments 
is increased to twelve. This article, which may be had in cither boxwood, ivory or plated metal, and of 
6, 12, 18 or 24 inches in length, is probably the most desirable for general use of any sold. 

251. In Fig. 123 we show what is known as a fiat scale, 
and which is also manufactured in both boxwood and ivory. 
Less scales or divisions can be put upon it than upon the trian- 
gular scale, yet for certain purposes it is to be preferred to the 
latter. There are less divisions to perplex the eye in hunting 
out just what is required, and, accordingly, there is less liability 
Fig. 124.— Flat Scale, with Dimensions of the Circle on the to error in its use. However, the limited number of scales 
Margins. which it contains greatly restricts its usefulness. 

252. In Fig. 124 we show another form of the fiat scale, one in quite common use in the past, but now 
virtually discarded in favor of more convenient dimensions and shapes. ' This scale combines with the various 
divisions of an inch the divisions of the protractor, as shown around the margin. The fact that the divisions 
of an inch for purposes of a scale are located in the middle of the instrument, away from the edge, which makes 
it necessary to step off all spaces for measurement with the dividers, renders the article awkward for use. The 
arrangement of the divisions of the circle, as shown on the margins, is less satisfactory for use than the same 
thing upon the circular protractor. 




u 



Drawing Tools and Materials. 



253. Lead Pencils. — Yarions qualities of pencils are sold, some at much lower prices than others, but, all 
thino-s considered, in this as in other cases, the best are the cheapest. Of leading brands, which are likely 
to give both draftsman and mechanic entire satisfaction, there may be mentioned Faber's, the American, 
and Dixon's. The former are perhaps the best known, having been before the public for the longest time, and 
accordingly we base our remarks concerning hardness, etc., upon them, as being likely to be more generally 
understood than if we referred to newer and less generally known pencils, although equally good. Both Faber's 
and the American, in the ordinary grades, employ numbers, 1, 2, 3, etc., to indicate hardness of lead, No. 1 
beinc the softest, and No. 5 being the hardest in common use. A finer grade of pencils manufactured by Faber, 
known as poliorades, is marked by letters, commencing at the softest with B B, and ending at the hardest with 
II II II H II H. The Dixon pencils -are graded to correspond with the qualities in greatest demand of the 
older manufacturers, but are marked upon a system peculiar to themselves. 



Side. 




Face. 



Fig. 125. — Pencil Sharpened to Chisel Point. 
254. Of either make of pencil the draftsman has the choice of round or hexagon shape, in all except the 
finest grades, the latter being made exclusively hexagon. The same quality of lead is said to be put in each, 
the only difference being in the shape of the wood and the finish. The round costs from 10 to 15 cents per 
dozen less than the hexagon. The poligradcs in price are about double the common pencils, and save for 
exceptionally fine work, which will be mentioned further on, are no better for the purposes of drawing and pat- 
tern cutting than the ordinary kind. Besides pencils with fixed leads, which we have been describing, there are 
several styles of pencils with movable leads. They are of various lengths and prices. Some are made of wood, 
hexagon in shape, finished and polished the same as an ordinary pencil, the point being of plated metal and 
the top surmounted by an ivory cap ; some are of hard rubber ; some are made of ivory. Leads of various 
qualities, and of different degrees of hardness, may be bought for any of them. While, no doubt, such articles 
are a trifle more ornamental than common pencils with fixed leads, we think that all gained in this direction is 
sacrificed in utility. The ordinary pencil is not only cheaper, but it is better for all practical purposes. 



Fig. 126.— Pencil Sharpened to Bound Point. Fig. 127.— Drawing Pen. 

255. "Whatever kind of pencil the draftsman or mechanic uses, he will require different numbers for differ- 
ent purposes. For working drawings, full-sized details, etc., on manila paper, a No. 3 is quite satisfactory. 
Some like a little harder lead, and therefore prefer a No. 4. For lettering and writing in connection with 
drawings upon manila or ordinary detail paper, a No. 2 is usually chosen. For fine lines, as in developing a 
miter, in which the greatest possible accuracy is required, a No. 5 is very generally used, although many pattern 
cutters prefer the finer grade for this purpose and use aHHH II H of the poligrades. 

256. The quality and accuracy of drawings depend, in a considerable measure, upon the manner in which 
pencils are sharpened. A pencil used for making straight lines, as, for instance, the measuring lines in miter 
cutting, and also in the dropping of points in pattern cutting generally, should be sharpened to a chisel edge, as 
illustrated in Fig 125. Pencils for making dots, for marking points and for general work away from the edges of 
the T-square, triangle, etc., should be sharpened to a round point, as shown in Fig. 126. It facilitates work, and 
it is quite economical to have several pencils at command, sharpened in different ways for different purposes. 
Where for any reason only one pencil of a kind can be had, both ends may be sharpened, one to a chisel edge 
and the other to a point. 

257. For keeping a good point upon a pencil, a piece of fine sand paper or emery paper, glued upon a piece 
of wood, will be found very serviceable. A flat file, mill-saw cut, is also useful for the same purpose. Sharpen 
the pencil with a knife, so far as the wood part is concerned, and then shape the lead as required upon the file 
or sand paper. 

258. Drawing Dens. — Although most of the pattern cutter's work is done by use of the pencil, there occa- 
sionally arise circumstances under which the use of ink is desirable. Tracings of parts of drawings are fre- 
quently required which can be better made with ink than with pencil. The pattern cutter, by the very force of 



D 



rawing 



Tools and Materials. 



25 



;?•■ 



.1 



:»' 



'# 



circumstances, gradually assumes the functions and duties of a draftsman, dependent altogether upon his skill 
in the management of tools, and his acquirement of knowledge concerning the draftsman's art. Therefore our 
remarks concerning drawing instruments would be quite incomplete with no mention of ink-iising tools and the 
management of ink itself. 

259. The drawing pen, as illustrated in Fig. 127, is used for drawing straight lines. Attachments with cor- 
responding members, to which the following remarks may also be applied, have been shown in connection with 
both compasses and beam compasses for drawing curved lines. The drawing j>en consists of two blades with 
steel points, fixed to a handle. The blades are so bent that a sufficient cavity is left between them for ink when 
the ends of the points meet close together or nearly so. The blades are set with the points more or less nearly 
together, by means of the screw shown in the engraving, so as to draw lines of any required thickness. One of 
the blades is provided with a joint, so that, by taking out the screw, the blades may be completely opened and the 
points readily cleaned after use. The ink is put between the blades by means of a common pen, or some- 
times by a small hair brash. In using the pen, it should be slightly inclined in the direction of the line to be 
drawn, and care must be taken that both points touch the paper. The drawing pen should be kept close to the 
ruler or straight-edge during the whole operation of drawing a line. 

260. To keep the blades of his pens clean is the first duty of a draftsman who is to make a good piece of 
work. Pieces of blotting, or unsized paper or cotton velvet, or even the 
sleeve of a coat, should always be at hand when a drawing is being inked. 
When a small piece of blotting paper is folded twice, so as to present 
a corner, it may be passed between the blades of the pen now and 
then, as the ink is liable to deposit at the point and obstruct the passage. 
To do this the screw must be loosened. The same purpose may be accom- 
plished, in a measure, by drawing the pen over a piece of velvet, or even 
over the surface of thick blotting paper. When the pen is done with for 
the occasion, it should be thoroughly cleaned at the nibs. This will pre- 
serve its edges and prevent rusting. If the draftsman is careless in this 
particular, the ink will soon corrode the points to such an extent that it will 
be impossible to draw fine lines. 

2G1. Pens will gradually wear away, and in course of time they require 
dressing. To dress up the tips of the blades of a pen, since they are gen- 
erally worn unequally by customary usage, is a matter of some nicety. A 
small oil stone is most convenient for use in the operation. The points 
should be screwed into contact in the first place, and passed along the stone, 
turning upon the point in a directly perpendicular plane until they acquire 
an identical profile. Next they are to be unscrewed and examined to ascer- 
tain the parts of unequal thickness around the nib. The blades are then 
to be laid separately upon their backs upon the stone, and rubbed down at 
the points until they are brought up to an edge of uniform fineness. It is 
well to screw them together again and pass them over the stone once or 
twice more to bring up any fault, to retouch them also at the outer and 

inner side of each blade to remove barbs or frasing, and finally to draw them Qualities. 

across the palm of the hand. 

202. India Ink. — For tracings, and for some kinds of drawings, which the pattern cutter is obliged to 
make occasionally, India ink is much better than the pencil, which is used for the greater part of his work. 
Care is to be exercised in the selection of ink, as poor grades are sold as well as good ones. Some little skill is 
required in dissolving or mixing it for use. 

263. India ink is sold in cakes or sticks, of a variety of shapes. It is prepared for use by the process tech- 
nically known as rubbing, which consists of dissolving a portion of it by rubbing it upon the surface of a glass, 
or of a porcelain slab or dish, in a very small quantity of water. 

264. As to the quality of the ink, upon general principles it may be determined by the price. The com- 



- 




Inferior. 
Figs. 12S and 129- 



Good. 
India Ink of Different 



mon size sticks are about 3 inches long. Inferior grades of this size can be bought at 40 cents, 50 cents and 60 



cents per stick, while good quality is worth $1.50 to $2 per stick, and the very best, still higher figures. How- 
ever, except in the hands of a responsible and experienced dealer, this method of judging is hardly satisfactory. 



26 



Draining Tools and Materials. 




Elevation, Cover on. 



To a certain extent ink may be judged by the brands npon it, although in the ease of the higher qualities the 
brands frequently change, so that this test may not be infallible. A common brand of ordinary quality (about 
50 cents per stick at present prices) is shown by our engraving, Fig. 128, full size. In shape the stick is oval, 
and is known as the " Lion's Head." An article of good quality for general use, and which is also adapted to 
fine work, is shown full sjze in Fig. 129. This stick is nearly square in shape, and at present price is worth $2. 
There is a great deal more ink in this stick than in the one first described, while its quality renders its use so 
much preferable to the other that it may safely be considered the cheaper of the two. These two brands have 
been selected for our illustrations because they are commonly known to the trade, and because they represent 
the two extremes between which the draftsman ordinarily chooses. There are other brands of about the same 
grade as each of these, and also those of intermediate and still better quality. 

265. The quality of India ink is quite apparent the moment it is used. The best is entirely free from grit 

and sediment, is not musky, and has a soft feel when wet- 
ted and smoothed. The color of the lines may also be 
used as a test of quality. With a poor ink it is impossi- 
ble to make a black hue. It will be brown or irregular 
in color. With poor ink the line will also present an ir- 
regular edge, as though broken or ragged, while an ink 
of satisfactory quality will produce a clean line, whether 
drawn very fine or quite coarse. 

266. In rubbing down ink ready for using, it should 
be made just so thick as to run freely from the pen. 
The degree can be determined at first by trial, but after 
a time it will be recognized by the appearance of the 
ink in the dish. The rubbing of a stick of ink in water 
tends to crack and break away the surface at the points. 
To prevent this, the stick may be shifted in the hand at 
intervals while being rubbed, thus rounding the surface. 
For the same reason, it is not advisable to bear very hard 
upon the stick while rubbing, as the mixture is otherwise 
more evenly made and the enamel of the pallet is less 
liable to be worn off. When drawings are being made 
which require the use of ink for some time, a considera- 
ble quantity of it should be rubbed down at one time, as the water continually evaporates. By having quite a 
quantity prepared, it will remain longer in fit condition for use. As evaporation takes place the ink may be 
thinned from time to time, as required, by the addition of more water. 

267. Various shaped cups, slabs and dishes are in use for mixing and containing India ink 
they are like those used for mixing and holding water colors. Indeed, in many cases the 
same articles are employed. Our engraving (Fig. 130) shows what is termed an India 
ink slab, with three holes and one slant. This article is in common use among drafts- 
men and serves a satisfactory purpose. In order to retard evaporation, a kind of sau- 
cers, in sets, is frequently used, so constructed that one piece will form a cover to the 
other, and which are known in the trade as cabinet sets or cabinet saucers. They are 
from 2^- to 3^ inches in diameter and come six in a set. In the absence of ware espe- 
cially designed for the purpose, India ink can be satisfactorily mixed in and used from 
an ordinary saucer or plate of small size, or even on a piece of glass. The articles made especially for it, however. 
are convenient, and in facilitating the care and economical use of the ink are well worth the small price they cost 

268. Thumb Tacks or Drawing Pins, both names being in common use, are made of a variety of sizes, 
ranging from those with heads one-quarter of an inch in diameter up to eleven-sixteenths of an inch in diameter, 
They are likewise to be had of various grades and qualities. The best for general use are those of German sil- 
ver, about three eighths to five-eighths of an inch in diameter, and with steel points screwed in and riveted 
Those which have the points riveted only, are of the second quality. The heads should be flat, to allow the 
J-square to pass over them readily. In the annexed cut, Fig. 131, we show an assortment of sizes. Those 
which are beveled upon their upper edges are preferable to those which are beveled underneath. 




Plan, View with Cover off. 
Fig. 130. — India Ink Slab with Cover. 



In many respects 




Fig. 131. — Thumb Tacks or 
Draiving Pins. 



Drawing Tools and Materials. 



27 




269. A Box of Instruments. — In Fig. 132 we show a box of instruments of medium grade, as made up 
and sold by the trade generally. While it contains some pieces that the pattern cutter has no use for, it also 
contains the principal tools he requires, all put together in compact shape, and in a convenient manner for 
keeping the instruments clean and in good order. The tray of the box lifts out, there being a space under- 
neath it in which may be placed odd tools, pencils, etc. "We do not recommend this particular box of instru- 
ments to the pattern cutter, nor, for that matter, any other. We introduce it as showing of what a box of tools 
ordinarily consists, and as indicating the advantages of a case in which to keep whatever tools the mechanic 
may possess. Tools may be selected, as required, of most of the large dealers in drawing instruments. A case 
or box fitted for their reception, neatly lined and with proper spaces, maybe obtained at a small additional cost. 
We believe it to be to the advantage of the pattern cutter to buy his instruments odd — that is, not to buy a case as 
ordinarily made up. By buying in single pieces he will get only what he requires for use, and will probably 
secure quite as good quality in the tools. After he has made his selection, a box properly fitted and lined 
should be provided for them. A little skill and ingenuity upon the part of the mechanic will enable him to 
make his own instrument case. Wood, as a material, is to lie preferred to metal, although there is less objection 
to the latter if the spaces for the instruments are properly padded 
and lined so that the tools need not come in contact with the metal 
of the box. Yelvet is probably the best material for the lining. 

270. India Rubber. — A good rubber Avith which to erase 
erroneous lines is indispensable in the pattern cutter's outfit. The 
several pencil manufacturers have put their brands upon rubber 
as well as upon pencils, and satisfactory quality can he had from 
any of them. In size, a large piece, since it continually be- 
comes less by use, is more economical than a small piece. The 
shape is somewhat a matter of choice. Flat cakes are perhaps 
the most used. The same quality can be obtained in diamond or 
lozenge shape, and in short square sticks or blocks. A very soft 
rubber is not so well adapted to erasing on detail paper as the 
harder varieties, but is to be preferred for use in fine drawings on 
holders are not economical for use upon rough paper, as they wear out too fast in such work. 

- 271. Besides the cakes aud blocks of rubber described above, rubber is fastened to pencils by a number of 
devices. There is the plain rubber cap ; the rubber let into the pencil, something as the lead is put in, and the 
rubber held in a metallic case, which also forms a shield for the point of the pencil when carried in the pocket. 
Rubber in all of these shapes is very useful and convenient, but considering the small quantity that can be 
got into any one of them, it should not be depended upon for other than occasional use where very small 
erasures are to be made. The larger piece, in the form of the cake first described, should be used for general 
work, and the piece in connection with the pencil used only as supplementary to it. 

272. Peeper. — The principal paper that the pattern cutter has anything to do with is known as brown detail 
paper, or manila detail paper. It can be bought of almost any width, from 30 inches up to 54 inches, in rolls 
of 50 to 100 pounds each. It is ordinarily sold in the roll by the pound, but can be bought at retail by the 
yard, although at a higher figure. There are different thicknesses of the same quality. Some dealers indicate 
them by arbitrary marks, as XX, XXX, XXXX; others by numbers 1, 2, 3; and still others as thin, medium 
and thick. The most desirable paper for the pattern cutter's use is one which combines several good qualities. 
It should be just as thin as is consistent with strength. A thick paper, like a stiff card, breaks when folded or 
Dent short, and is, therefore, objectionable. The paper should be very strong and tough, as the requirements in 
use are quite severe. The surface should be very even and smooth, yet not so glossy as to be unsuited to the 
use of hard pencils. It should be hard, rather than soft, and should be of such a texture as to withstand 
repeated erasures in the same spot without damage to the surface. 

273. White drawing paper, which the pattern cutter has occasionally to use in connection with his work, 
can be had of almost every conceivable grade and in a variety of sizes. The very best quality, and the kinds 
suited for the finest drawings, come in sheets exclusively, although the cheaper kinds are also made in the shape 
of sheets as well as in rolls. White drawing paper in rolls can be bought of different widths, ranging from 36 to 
54 inches, and from a very thin grade up to a very heavy article, and of various surfaces. It is sold by the 
pound, rolls ranging from 30 to -±0 pounds each, and also at retail by the yard. 



Fig. 132. — A Box of Instruments. 

d quality paper. Erasers put up in wooden 



28 Drawing Tools and Materials. 

274. Drawing paper in sheets is sold by the quire, and at retail by the single sheet. The sizes are generally 
indicated by names which have been applied to them. The following are some of the terms in common use, 
with the dimensions which they represent placed opposite : 



Cap 13x17 

Demy 15 x 20 

Medium 17 x 22 

Eoyal 19x24 



Columbier 


. . 23 x 35 


Double Elephant . . 


. . 27 x 40 


Antiquarian 


. . 31 x 53 


Emperor 


. . 48 x GS 



Super Eoyal 19 x 27 

Imperial." . 22 x 30 

Elephant 23 x 28 

Atlas 20 x 34 

Still another set of terms is used in designating French drawing papers. Different qualities of paper, both as 
regards thickness, texture and surface, can be had of any of the sizes above named. 

275. The pattern cutter has frequent use for tracing paper, and a good article, one which combines strength, 
transparency and suitable surface, is very desirable. Tracing paper is sold both in sheets, in size to correspond 
to the drawing papers above described, and in rolls, to correspond in width to the roll drawing paper. It is 
usually priced by the quire and by the roll, although single sheets or single yards are to be obtained at retail. 
The rolls, according to the kinds, contain from 20 to 30 yards. "We cannot offer any other good rule for selec- 
tion of suitable quality than inspection and actual trial. There are various manufacturers of this article, but it 
is usually sold upon its merits, rather than by any brand or trade-mark. Tracing cloth, or tracing linen, is used 
in place of tracing paper where great strength and durability are required. This article comes exclusively 
in rolls, ranging in width from IS to 42 inches. There are generally 24 yards to the roll, and prices are made 
according to the width, or, in other words, according to the superficial contents of the roll. Two grades are 
usually sold, the first being glazed on both sides and suitable only for ink work, and the second on but one side, 
the other being left dull, rendering it suitable for pencil marks. Upon general principles, pencil marks are 
not satisfactory upon cloth, even upon the quality specially prepared with reference to them. It is but a very 
little more labor or expense to use ink, and a much more presentable and usable drawing is made. Tracing 
paper may be used satisfactorily with either pencil or pen. 



GEOMETRICAL PROBLEMS. 



276. Very much of the pattern cutter's skill depends upon his knowledge of fundamental geometrical prin- 
ciples. He should know how to lay off an octagon or a pentagon, or any required figure or angle, as well as 
how to cut a miter to fit the given angle after the figure is drawn. He should know how to draw a plan and 
elevation of an oval flaring dish, as well as how to develop the patterns for it after the drawing is given him. 
It is designed that this book shall be complete in itself — that it shall fully illustrate the science and art of pattern 
cutting in all its phases. It does not presuppose a knowledge of geometry upon the part of the student, but 
undertakes to supply all that it is necessary for him to learn in acquiring a knowledge of pattern cutting, from 
the definition of simple terms, up to the cutting of the most intricate and complex patterns. In the preceding 
chapters we have described at some length the various instruments and tools which the pattern cutter is likely 
to use, and have, along with other terms, defined and illustrated various figures and shapes in which his work is 
likely to occur. It remains for us, therefore, to illustrate the use of these instruments and tools, and 
show methods of constructing the various figures commonly occurring in pattern cutting, before commencing 
the demonstration of practical problems. There are also various expedients for shortening and simplifying 
what would otherwise prove long and tedious operations, at which it will be well to glance in passing. In the 
arrangement of the problems in this chapter, it has been found difficult to follow any one logical system through- 
out. Several schemes of order for the problems have suggested themselves, each of which, for certain parts, 
has appeared better than the others. Accordingly, to the critical reader, the arrangement as here presented may 
appear defective in some particidars, or, at least, show inconsistencies. But the student is reminded that the intent 
of the book is to afford not only a complete exposition of the art of pattern cutting, but also to serve as a ready 
reference book for answering vexed questions. It attempts not only to present the subject, from beginning 
to end, in an arrangement that will be acceptable to those who desire to make the book a regular and systematic 
study, but also to exhibit each individual principle and problem in a complete and independent form, so that 
when any one item is referred to it shall be found self-explanatory, and therefore ready for use, without tedious 
search through problems in other portions of the book in order to fully comprehend it. Accordingly, the use 
of the index is recommended to all who desire to pursue any order of study different from the arrangement 

we have followed. Since each rule and demonstration, so far as possible, - 

is made independent of all other rules and demonstrations, the student, *""" 

by referring to the several pages indicated by the topic heads as given in 

the index, can obtain an exhaustive presentation of any phase of the sub- | 

ject upon any system of classification he chooses to follow. "We deem no 

further explanation necessary for the somewhat arbitrary arrangement we 

have found it desirable to follow in carrying out the special purposes of -c l 



y 



the book. ^ pifJ _ r33 _ To Draw a strai ght Line Par- 

277. To Drain a Straight Line Parallel to a Given Line, and at a a iiei to a Given straight Line, and at a 

Given Distance from it, Using the Compasses and a Straight-Edge. — In Given Distance from it, Using the Com- 

Fig. 133, let C D be the given line, parallel to which it is desired to draw P asses and a straight-Edge. 

another straight line. Take any two points, as A and B, in the given line as centers, and, with a radius equal 
to the given distance, describe the arcs x x and y y. Draw a line touching these arcs, as shown by E F. Then 
E F will be parallel to C D. 



30 



Geometrical Problems. 




Fig. 134. — To Draw a Line Parallel to 
Another by the Use of Triangles or 
Set-Squares. 



other shall be in a direction 
accuracy is attainable in this 



\ 



\ 



^ 



Fig. 135. — To Erect a Perpendicular at 
a Given Point in a Straight Line by 
Means of the (Compasses and Straight- 
Edge. 



278. To Draw a Line Parallel to Another by the Use of Triangles or Set-Squares. — In Fig. 134, let A B be 
the line parallel to which it is desired to draw another. Place one of two triangles or set-squares, F 1 , against it, as 
indicated by the dotted lines. While holding F 1 firmly in this position, bring the second triangle, E, against one 

of its other sides, as shown. Then, holding the second triangle firmly in 
place, slide the first away from the given line, keeping the edges of the two 
triangles in contact, as shown in the figure. Against the same edge of the 
first triangle that was placed against the given line draw a second line, as 
shown by C D. Then C D will be parallel to A B. In drawing parallel 
lines by this method, it is found advantageous to place the longest edges of 
the triangles against each other, and to so place the two instruments that 
the movement of one triangle against the 
oblique to the lines to be drawn. Greater 
way than is possible otherwise. 

279. To Erect a Perpendicular at a Given Point in a Straight Line by 
Means of the Compasses, and Straight-Edge. — In Fig. 135, let A B represent 
the given straight line, at the point C in which it is required to erect a per- 
pendicular. From C set off on each side equal distances of any convenient 
space, as shown by D and B. "With D and B as centers, and with any radius 
longer than the distance from each of these points to C, strike arcs, as shown 
by x x and y y. From the point at which these arcs intersect, E, draw a - 
line to the point C, as shown. Then E C will be perpendicular to A B. 

280. To Erect a Perpendicular at or near 
the End of a Given Straight Line by Means 
of the Compasses and Straight-Edge. — First 

Method. — In Fig. 136, let A B be the given straight line, to which, at the point 
P, situated near the end, it is required to erect a perpendicular. Take any 
point (C) outside of the line A B. With C as center, and with a radius equal to 
the distance from C to P, strike the arc, as shown, cutting the given line A B in 
the point P, and also in another point, as at E. From E, through the center C, 
draw the line E F, cutting the arc, as shown at F. Then from the point F, thus 
determined, draw a line to P, as shown. The line F P is perpendicular to A B. 

281. To Erect a Perpendicular at or near # 
the End of a Given Straight Line by Means D / 
of the Compasses and Straight- Edge. — Second 
Method,— In Fig. 137, let B A be the given 

straight line, to which, at the point P, it is required to erect a perpendicular. 

From the point P, with a radius equal to three parts, by any scale, describe an 

arc, as indicated by x x. From the same point, with a radius equal to four 

parts, cut the line B A in the point C. From the point C, with a radius equal 

to five parts, intersect the arc first drawn by the arc y y. From the point of 

intersection D draw the line D P. Then D P will be perpendicular to B A. 
282. To Draw a Line Perpendicular to Another Line by the Use of Tri- 
angles or SeLSguares. — In Fig. 138, let C D be 
the given line, perpendicular to which it is 
required to draw another line. Place one side 

of a triangle, B, against the given line, as shown. Bring another triangle, A, or 
any straight edge, against the long side of the triangle B, as shown. Then 
move the triangle B along the straight-edge or triangle A, as indicated by the 
dotted lines, until the opposite side of B crosses the line C D at the required 
point. When against it, draw the line E F, as shown. Then E F is perpen- 
dicular to C D. It is evident that this rule is adapted to drawing perpendic- 
ulars at any point in the given line, whether central or located near the end. Its use will be found especially 
convenient for erecting perpendiculars to lines which run oblique to the sides of the drawing board. 



\ 



/ 
/ 

Kb 



A 



\ 



/ 



/ 



Fig. 136. — To Erect a Perpendicular 
at or near the End of a Given 
Straight Line by Means of the 
Compasses and Straight-Edge. — 
First Method. 



X 



V 






\" 



n. 



% 



X 



\ 




P 4 Paris C 

Fig. 137. — To Erect a Perpendicular 
at or near the End of a Given 
Straight Line by Means of the 
Compasses and Straight-Edge. — ■ 
Second Method. 



Fig. 138. — To Draw a Line Perpen- 
dicular to Another by the Use of 
Triangles or Set-Squares. 



Geometrical Proble 



HIS. 



31 



a \ 


?A 






/ 


\ 


/ 


\ 


/ 


\ 


/ 


\ 


/ 


\ 


/ 


\ 


/ 


\ 


/ 

/ 

/ 

/ 
1 

1 


\ 

\ 
\ 

\ 
1 
1 


1 

1 

\ 

\ 

\ 

\ 

\ 


1 
1 
/ 
/ 

/ 
/ 
/ 

/ 


\ 


/ 


\ 


/ 


\ 


/ 


\ 


/ 












\ 


e-H'f 




Fig. 139. — To Divide, a Given Straight Line 
into Two Equal Parts, with the Com- 
passes, by Means of Arcs. 



2S3. To Divide a Given Straight Line into Two Equal Parts, with the Compasses, by Means of Arcs. In 

Fig.139, let it be required to divide the straight line A B into two equal parts. From the extremes A and B as 
centers, and with any radius greater than oue-half of A B, describe the arcs 
df and a e, intersecting each other ou opposite sides of the given line A B. 
A line drawn through these points, as shown by G II, will bisect the line 
A B, or, in other words, divide it into two equal parts. 

284. To Divide a Straight Line into Two Equal Parts by the Use of 
a pair of Dividers. — In Fig. 140, it is required to divide the line A B into 
two equal parts, or to find its middle point C. Open the dividers to as near 

h-df of the given line as possible by the eye. Place one point of the divid- ^ 

ers on one end of the line, as at A. Bring the other point of the dividers 
to the line, as at C, and turn on this point, carrying the first around to D. 
Should the point D coincide with the other end of the line, we have the 
division required. But should the point D fall within (or without) the 
end of the line, divide this deficit (or excess) by the eye into two equal 

parts, and extend (or contract) 
the opening of the dividers to 
this point and apply them again 
as at first. Thus, finding that 

the point D falls within the end of the line, we know our first division 
is too short. We therefore divide the deficit D B by the eye, as shown 
by E, and increase the space of the dividers to the amount of one of 
these divisions. Then, commencing again at A, we step off as before, 
and finding that upon turning the dividers upon the point F the point coincides with, the end of the line B, we 
know that F is the middle point in the line. In some cases it may be necessary to repeat this operation several 
times before the exact center is obtained. The smaller the space to be divided, the more accurate is the S23acino- 
of it by the eye. 

285. To Divide a 'Straight Line into Tioo Equal Parts by 
ilie Use of a Triangle or Set-Square. — In Fig. 141, let A B be a 
given straight line. Place a J-square or some straight edge parallel 
to A B. Then bring one of the right-angled sides of a set-square 
against it, and slide it along until its long side, or hypothenuse, 
meets one end of the line, as A. Scribe along the long side of 
the triangle indefinitely. Peverse the position of the set-square, 
as shown by the dotted lines, bringing its long side against the 
end, B, of the given straight line, and in like manner scribe along 
its long side. Next slide the set-square along until its vertical side 
meets the intersection of the two lines scribed, as shown at C, from 
which point drop a perpendicular to the line A B, cutting it atD. 
Then D will be equidistant from the two extremities A and B. 

286. To Divide a Given Straight Line into Any Number 
of Equal Parts. — In Fig. 142, let AB be a given straight line to 

„ B = I* c" 



Fig. 140. — To Divide a Line into Two Equal 
Parts by the Dividers. 




Fig. 141. — To Divide a Straight Line into Two Equal 
Parts by the Use of a Triangle or Set-Square. 

be divided into equal parts, in this case eight. From one extremity of this 
line, as at A, draw a line, as either A C or A D, oblique to A B. Set the 
dividers to any convenient space, and step off the oblique line, as A C, eight 
divisions, as shown by a I d, etc. From the last of the points, h, thus ob- 
tained, draw a line to the end of the given line, as shown by h h\ Parallel 
to this line draw other lines, from each of the other points to the given line. 
The divisions thus obtained, indicated in the engraving by «'" b~ c 2 , etc., will 
be the desired spaces in the given line. It is evident by this rule that it is 
convenience, at what space the dividers are set. The object of the second 
oblique line in the engraving is to illustrate this. Upon A C the dividers were set so as to produce spaces 
shorter than those required in the given line A B, while in A D the spaces were made longer than those 




A 

Fig. 142. — To Divide a Given Straight 
Line into Any Number of Equal Parts. 

immaterial, except as a matter of 



Geometrical Problems. 




Fig. 143.—4 Scale by which to Divide a Straight Line into Any Number of Equal Parts. 



required in the given line. By connecting the extremes, as shown by the lines h h' and A 1 h\ and drawing lines 
from the points in each line parallel to these lines respectively, it will be seen that the same divisions are 
obtained in the given line A B. 

287. A Scale by which to Dwide a Straight Line into Any Number of Equal Parts.— -It frequently 'happens 
in pattern cutting that it is more convenient to transfer the length of a given line to a slip of paper, and by laying 

the paper across a scale, as shown in 
Fig. 113, mark the required dimen- 
sions upon it, and afterward transfer 
them to the given line, than to di- 
vide the line itself by one of the 
methods explained for that purpose. 
It also occasionally occurs that it is 
desirable to divide lines of different 
lengths into the same number of 
equal parts, or the same lengths of 
lines into different numbers of equal 
parts. Such a scale as is shown in 
Fig. 113 is adapted to all of these 
purposes. The scale may be ruled 
upon a piece of paper or upon a 
sheet of metal, as is preferred. The 
lines may be all of one color, or 
two or more colors may be alter- 
nated in order to facilitate counting the lines or following them by the eye across the sheet. In size, the scale 
is to be adapted to the special purposes for which it is intended to be used. For cornice makers' use it should 
not be less than 18 inches in width, and might with advantage be as wide as the widest sheet of metal commonly 
worked. The length should be proportioned to the width, to adapt it to the use of strips diagonally, as shown 
in the eno-ravincr. The size of the spaces into which it is to be divided also depends altogether upon the char- 
acter of the work in connection with which it is to be used. For cornice makers' purposes, the divisions might 
be made from a half inch up to an inch in width. By the contrast of two colors in ruling the lines, one 
scale may be adapted to both coarse and fine work. For instance, if the lines are ruled a half inch apart, in 
colors alternating red and blue, in fine work all the lines in a given space may be used, while in large work, in 
which the dimensions are not required to be so small, either all the red or all the blue lines may be used, to the 
exclusion of those of the other color. We have indicated approximately the size desirable in such a scale for 
cornice makers' use. "When designed for other purposes, the size must be made suitable. In Fig. 143, let it be 
required to divide the line A B into thirty equal parts. Transfer the length A B to a slip of paper, as shown by 
A 1 B 1 and placing A 1 against the first line of the scale, carry B 1 to the thirtieth line. Then 
mark divisions upon the strip of paper opposite each of the several lines it crosses, as shown. 
Let it be required to divide the same length, A B, into fifteen equal parts by the scale. 
Transfer the length A B to a straight strip of paper, as shown by A 3 B\ Place A 3 against 
the first line and carry W against the fifteenth line, as shown. Then mark divisions upon 
the strip of paper opposite each line of the scale, as shown. A problem of frequent occur- 
rence in pattern cutting is to divide the circumference of a circle into a given number of 
equal parts. By first obtaining a straight line equal to the circumference of the circle, the 
division may be readily performed by means of this scale. Several rules for obtaining a 
straight line approximately equal to the circumference of a circle are given in their appro- 
priate place. 

288. To Divide a Given Angle into Two Equal Parts.— In Fig. 144, let A C B repre- 
sent any angle, through the center of which it is required to draw a straight line. From the 
vertex, or point C, as center, with any convenient radius, strike the arc D E. From D and E 
as centers, with any radius greater than one-half the length of the arc D E, strike short arcs intersecting at G, as 
shown. Through the point of intersection, G, draw a line to the vertex of the angle, as shown by F C. Then 
F C will divide the angle into two equal parts. 




Fig. 144. — To Divide 
a Given Angle into 
Two Equal Parts. 



Geometrical Problems. 



33 




Fig. 145. — To Find the Center from 
which a Given Arc is Struck. 



289. To Find the Center from which a Given Arc is Struck. — In Fig. 145, let A B C represent the <nven 
arc, the center from which it was struck being unknown and to be found. From any point near the middle of the 
arc, as B, with any convenient radius, strike the arc F G, as shown. Then from the 
points A and C, with the same radius, strike the intersecting arcs I H and E D. 
Through the points of intersection draw the lines K M and L M, which will meet 
in M. Thus M is the center from which the given arc was struck. Instead of 
the points A and C being taken at the extremities of the arc, which would be 
quite inconvenient in the case of a long arc, the points may be located in any 
part of the arc which is most convenient. The greater the distance between A 
and B, and B and C, the greater will be the accuracy of succeeding operations. 
The essential feature of this rule is to strike an arc from the middle one of the 
points, and then strike intersecting arcs from the other two points, using the 
same radius. It is not necessary that the distance from A to B and from B to 
C shall be exactly the same. 

290. The Chord and /light of a Segment of a Circle being Given, to Find 
the Center by which the Arc may be Struck. — In Fig. 146, let A B represent the 
chord of a segment or arc of a circle, and D C the rise or night. It is required to 

find a center from which an arc, if struck, will pass through the three points A, D 
and B. Draw A D and B D. Bisect A D, as shown, and prolong the Hue H L indefi- 
nitely. Bisect D B and prolong I M until it cuts H L, produced in the point E. 
Then E, the point of intersection, will be the center sought. It will be observed 
that by producing D C, and intersecting it by either HLorll prolonged, the 
same point is found. Therefore, if preferred, the bisecting of either A D or D B 
may be dispensed with. A practical application of this rule occurs quite frequently 
in cornice work, in the construction of window caps and other similar forms, to fit 
frames already made. In the conveying of orders from the master builder or 
carpenter to the cornice worker, it is quite customary to describe the shape of the 
head of the frames which the caps are to fit by stating that the width is, for exam- 
ple, 36 inches, and that the rise is 4 inches. To draw the shape thus described, 
proceed as follows : set 
off A B equal to 36 
inches, from the center 

of which erect a perpendicular, D C, which make 

equal to 4 inches. Continue D C in the direction of 

E indefinitely. Draw A D, which bisect, as shown, 

and draw II L, producing it until it cuts D C pro- 
longed, in the point E. Then with E as center 

and E D as radius, strike the arc A D B. 

291 To Find the Center from which a Given 

Arc is Struck by the Use of the Square. — In Fig. 

147, let A B C be the given arc. Establish the 

point B at pleasure and draw two chords, as shown 

by A B and B C. Bisect these chords, obtaining 

the points E and D. Place the square against the 

chord B C, as shown in the engraving, bringing the 

heel against the center point, D, and scribe along 

the blade indefinitely. Then place the square as 

shown by the dotted lines, with the heel against the 

center point, E, of the second chord, and in like 

manner scribe along the blade, cutting the first line 

in the point F. Then F will be the center of the 

circle, of which the arc A B C is a part. This rule 

will be found very convenient for use in all cases where the radius is less than twenty-four inches in length. 




Fig. 146. — The Chord and Flight 
of a Segment of a Circle be- 
ing Given, to Find the Center 
by which the Arc may be 
Struck. 




Fig. 147.- 



-To Find the Center from which a Given Arc is Struck by 
the Use of the Square. 



34 



Geometrical Problems. 






292. To Strike a Segment of a Circle by a Triangular Guide, the Chord and Right being Given. — In Fig. 
148, let A D be the given chord and B F the given bight. The first step is to determine the shape and size of the 
triangular guide. Connect A and F, as shown. From F, parallel to the given chord A D, draw F G, making 

it in length equal to A F, or longer. Then A F G, as shown in the en- 
graving, is the angle of the triangular guide to be used. Construct the 
guide of any suitable material, making the angle of two of its sides equal to 
the angle A F G. Drive pins at the points A, F and D. Place the guide 
as shown. Put a pencil at the point F. Shift the guide in such a manner 

Fig. i 4 8— To strike a Segment of a circle that the pencil will move toward A, keeping the guide at all times 
by a Triangular Guide, the chord and against the pins A and F. Then reversing, shift the guide so that the 
Hight being Given. pencil at the point F will move toward D, keeping the guide during this 

operation against the pins F and D. By this means the j^encil will be made 

to describe the arc A F D. 

293. To Draw a Circle Through any Three Given Points not in a 
Straight Line. — In Fig. 149, let A, D and E be any three given points not 
in a straight line, through which it is required to draw a circle. Connect 
the given points by drawing the lines A I) and D E. Bisect the line A D 
by F C, drawn perpendicular to it, as shown. Bisect D E by the line 
G C, also perpendicular to it, as shown. Then the point C, at which these 
lines meet, is the center of the required circle. 

294. To Raise a Perpendicular to an Arc of a Circle, without hav- 
ing Recourse to the Center. — In Fig. 150, 
let A D B be the arc of a circle to which 
it is required to erect a perpendicular. 
With A as center, and with any radius 
greater than half the length of the given 

arc, describe the arc x x, and with B as center, and with the same radius, 
describe the arc y y, intersecting the arc first struck, as shown. Through 
the points of intersection draw the line F E. Then F E will be perpen- 
dicular to the arc, and if sufficiently produced will reach the center from 
which the arc A B is drawn. 

295. To Drain a Tangent to a Circle, or a Portion of a Circle, without 

having Recourse to the Center. — In 

Fig. 151, let A D B be the arc of a 

circle, to which a tangent is to be 

drawn at the point D. With D as 
center, and with any convenient radius, describe the arc A F B, cutting 
arc in the points A A B. Join the points A and B, as 
shown. From D draw a straight line 
perpendicular to A B, as shown by 
D C, and from B erect another per- 
pendicular to A B, as shown by B G. 

Make B G equal to C D. Draw E H through the points D and G. 
E H will be the required tangent. 

296. To Draw a Straight Line Equal to the Circumference of a Given 
Circle.— First Method.— in Fig. 152, let A D B C be the circle, equal to the 
circumference of which it is desired to draw a straight line. Draw two diam- 
eters, A B and D C, as shown, at right angles. Connect the points A and D. 
Bisect the line A D and draw E F. To three times the diameter (A B or 
D C) add the length E F. The result will be very nearly the circumference 
of the circle. This rule gives a length slightly in excess of the true circum- 

Fig. 152. — To Draw a Straight Line „ . , . ° P , „ 

Equal to the Circumference of a ference, the error being about one-sixteenth of an inch in the circumference 
Given Circle.— First Method. of a circle the diameter of which is one foot. 



Fig. 149. — To Draw a Circle Through 
any Three Given Points not in a 
Straight Line. 



Fig. 150. — To Raise a Perpendicular to 
an Arc of a Circle, without having 
Recourse to the Center. 



the given 





Fig. 151. — To Draw 
or a Portion of a 
Recourse to the Center, 



a Tangent to a Circle, 
Circle, without having 



Then 



Geometrical Problems. 



35 




Fig. 153. — To Draw a Straight Line 
Equal to the Circumference rf a 
Given Circle. — Second Method. 




V9 

_.io\d 
F '"EC 

Fig. 154. — To Draiv a Straight Line 
Equal to the Semi-Circumference of 
a Given Circle. 



297. To Draw a Straight Line Equal to the Circumference of a Given Circle. — Second Method.— In Fi°\ 
153, let A D B C be the circle, equal to the circumference of which it is required to draw a straight line. Draw 

any two diameters at right angles, as shown by A B and I) 0. Divide one of 
the four arcs, as, for instance, D B, into eleven equal parts, as shown. From 
9, the second of these divisions from the point B, let fall a perpendicular to 
A B, as shown by 9 F. To three times the diameter of the circle (A B or D C) 
add the length 9 F, and the result will be a very close approximation to the 
length of the circumference. This rule, upon a diameter of 1 foot, gives a length 
of about -g%ths of an inch in excess of the actual length of the circumference. 

298. To Draw a Straight Line Equal 
to the Semi- Circumference of 'a Given Circle. 
— In Fig. 154, let A B C represent the semi- 
circle, equal to the circumference of which 
it is required to draw a straight line. Di- 
vide the semicircle into two equal parts by 
the line B F. Divide the arc B C into 
eleven equal parts, as shown by the small 
figures. From the first division, above the radius F C, drop a perpendicular 
upon that line, as shown by D E. To three times the radius F C add the 
distance D E. The result will be the length of the semi-circumference ABC. This rule, in principle, is the 
same as that presented in Section 295, to wdiich refer for the measure of its accuracy. 

299. To Draw « Straight Line Equal to the Quarter Circumference of 
a Given Circle. — In Fig. 155, ABC represents a quarter circle, equal to the 
arc A C of which it is required to draw a straight line. Divide the arc into 
eleven equal parts, as shown, and from the first division above the radius 
B C drop a perpendicular on to B C, as shown by D C. Bisect this perpen- 
dicular, as shown at F. Also bisect the radius B C, as shown at E. Then to 
three times B E, or one-half of the radius, add the length D F, being half 
of the perpendicular. The result will be the length of a straight line which 
shall equal the quarter circumference, or the arc A 0. This rule, also, is the 
same in principle as that given in Section 295, and produces a close approxi- 
mation to the actual length. 

300. To Draw a Straight Line Equal to 
any Given Part of a Circle less than a Semi- 
circle. — First Method, Using the Center. — In 
Fig. 156, let A F B represent the given arc, 
equal to which it is desired to draw a straight 
line. Draw the chord A B to the arc, which 

bisect. From the middle point C, through the center of the circle of which the 

given arc is a part, draw the line J K indefinitely. Divide a radius of the circle, as, 

for example, D E, into four equal parts, and 
set off three of those parts from E toward K, 
as indicated by the small figures. Draw the 
tangent G II to the arc at the point F, or 
where J K cuts the arc. From the point L, 
obtained as just before explained, draw lines 
through the extremities of the arc, or through 
A and B, cutting the tangent in the points G 
and H. The line G H will then be equal to Fig. 156 
the length of the arc A B. This rule, like Equal 
others of its class, is only approximately cor- 
rect, but the variation is so slight as to make 




Ffy- 155- — To Draw a Straight Line 
Equal to the Quarter Circumference 
of a Given Circle. 





E 
Fig. 157. — To Draw a Straight Line 
Equal to any Given Part of a Circle less 
than a Semicircle. — Second Method, 
Without Using the Center. 

its use entirely safe in ordinary mechanical operations. 

301. To Draw a Straight Line Equal to cony Given 



-To Draw a Straight Line 
to any Given Part of a 

Circle less than a Semicircle. — First 

Method, Using the Center. 



Part of a Circle less than a Semicircle. — Second 



36 



Geometrical Problems. 




Fig. 158. — To Draw an Ogee by Means 
of Two Quarter Circles. 



UJ 






G 


/ '' ' 




■^ 


^ 


K 


F 




B H 




A 



Method, Without Using the Center.— In Fig. 157, let A B C represent the given part of a circle, equal to which a 
straight line is to be drawn. Draw the chord A C, which bisect as shown by D E. Draw A B, which is the 

chord of half of the given arc. Lay off the length A B twice on the chord 
A C. The distance will exceed the length A C, as indicated by A G and G H, 
by a certain distance. Divide this excess, or the space C H, into three equal 
parts, and increase the length stepped off by twice the chord A B, by the 
amount of one of these parts, as shown by H D. Then A J will be a 
straight line, which in length is equal to the arc ABC. This rule, like 
the one preceding it, is sufficiently accurate for ordinary mechanical opera- 
tions, but is not absolutely correct. 

302. To Divide an Arc of a Circle into Any Given Number of 
Equal Parts. — Several somewhat intricate rules for performing this operation 

have been devised, but as they are not practical, we have not considered it 
worth while to present them in this connection. The simplest way of perform- 
ing this oft-recurring operation in the pattern cutter's work is as follows : Lay 
off a straight line equal to the are of the circle by either of the rules already 
given, by stepping around the arc with the dividers, or by measuring the arc 
with a strip of metal bent to fit it. Having obtained the straight line by one 
or the other of these ways, divide it into the required number of equal parts 
by either of the rules already given for that operation. Take one of the 
spaces thus obtained in the dividers, or, what is better for the purpose, the n 9- ^9-—To Draw a Grecian Ogee. 
spacers, and step around the arc, marking the places where the points of the instrument come. 

303. To Draw an Ogee by Ifeans of Two Quarter Circles. — In Fig. 
158, let A D be the bight of the ogee, which is also equal to D B, the 
projection. Bisect A D, obtaining the j>oint C, from which, parallel to 
D B, draw C G. From C as center, and with C A as radius, describe the 
arc A F. From B erect a perpendicular, cutting C G in the point G. 
From G as center, and with G F, which is equal to A C, as radius, strike 
the arc F B, which will complete the ogee. 

304. To Draw a Grecian Ogee. — In Fig. 159, let A D be the bight of 
the required form and A B the projection. Upon these two sides erect a 
rectangle, as shown by B A D E. 
Bisect A D, and through the point 
F thus obtained draw a line at right 
angles to A D indefinitely, as shown 
by M L. Bisect A B by the line 

H I, which make equal to A D, cutting ML in the point K. Make 
F L and G M each equal to IT F. Divide I D, D F, G B and B II 
into the same number of equal parts. From the divisions in I D and 
B H draw lines to K. From L draw lines through the points in D F, 
to intersect the lines drawn from I D, and from M, through the divis- 
ions in G B, draw lines to intersect the lines drawn from B II. A line 
traced through the points of intersection thus obtained will be the curve 
sought. 

305. To Draw a Parabola by the Intersection of Lines, its Sight and 
Base or Ordinate being Given. — In Fig. 160, let A B be the hight and 
D C the base of the required figure. Draw D E and C F equal to the hight 
and parallel to it. Divide D E and C F into any convenient number of 
equal parts. Divide each half of the base into the same number of Fi 9- 161.— To Draw a Simple Volute. 
equal parts, as shown. Draw lines from the points 1 2 3 4 in D E and C F to the point B. Erect perpendic- 
ulars to the base D C on each of the points 12 3 4. Then a line traced through the points in which these lines 
intersect will describe one-half of the required figure. 

306. To Draw a Simple Volute.— Let D A, in Fig. 161, be the width of a scroll or other member for which 
it is desired to draw a volute termination. Draw the line D 1, in length equal to three times D A, as shown by 




A •* 3 C 

Fig. 160. — To Draw a Parabola by the 
Intersection of Lines, its Hight and 
Base or Ordinate being Given. 







Geometrical Problems. 



37 



DA, AB and B 1. From the point 1 draw 1 2 at right angles to D 1, and in length equal to two-thirds the 
width of the scroll, or, what is the same, to two-thirds the width of D A. From 2 draw the line 2 3 perpen- 
dicular to 1 2, and in length equal to three-quarters of 
A D. Draw the diagonal line 1 3. From 2 draw a 
a line perpendicular to 1 3, as shown by 2 4, indefi- 
nitely. From 3 draw a line perpendicular to 2 3, pro- 
ducing it until it cuts the line 2 4 in the point 4. 
From 4 draw a line perpendicular to 3 4, producing it 
until it meets the line 1 3 in the point 5. In like 
manner draw 5 6 and 6 7. The points 1, 2, 3, 4, etc., 
thus obtained are the centers by which the curve of 
the volute is struck. From 1 as center, and with 1 D 
as radius, describe the quarter circle D C. Then from 
2 as center, and 2 C as radius, describe the quarter cir- 
cle G F, and so continue until the figure is completed, 
as shown. 

307. To Describe an Ionic Volute. — Draw the line 
A B, Fig. 162, equal to the hight of the required 
volute, and divide it into seven equal parts. From the 
third division draw the line 3 C, and from a point at 
any convenient distance on this line from A B describe 
a circle, the diameter of which shall equal one of the 
seven divisions of the line A B. This circle forms 
the eye of the volute. In order to show its dimen- 



sions, etc., it is enlarged in 



Fig. 



163. A square, 




Fig. 162. — To Describe an Ionic Volute. 



D E F G, is constructed, and the diagonals G E and 
F D are drawn. F E is bisected at the point 1, and the line 1 2 is drawn parallel to G E. The line 2 3 
is then drawn indefinitely from 2 parallel to F D, cutting G E in the point II. The distance from II to the 
center of the circle, 0, is divided into three equal parts, as shown by H a <b 0. The triangle 2 O 1 is formed. 

On the line O H set off a point, as c, at a distance from O equal to 
one-half of one of the three equal parts into which O H has been 
divided. From c draw the line c 3 parallel to 1 O, producing it 
until it cuts 2 3 in the point 3. From 3 draw the line 3 4 parallel 
to G E indefinitely. From the point c draw a line c 4 parallel to 
2 O, cutting the line 3 4 in the point 4, completing the triangle 
c 3 4. From 4 draw the line 4 5 parallel to F D, meeting 1 in 
the point 5. From 5 draw the line 5 6 parallel to G E, meeting the 
line 2 O in the point 6. From 6 draw the line 6 7 parallel to F D? 
meeting the line c 3 in the point 7. Proceed in this manner, obtain- 
ing the remaining points, 8, 9, 10, 11 and 12. These points form 
the centers by which the outer line of the volute proper is drawn. 
From 1 as center, and with radius 1 F 1 , describe the quarter circle 
F 1 G 1 . Then form 2 as center, and with radius 2 G 1 describe the 
quarter circle G 1 D 1 , and so continue striking a quarter circle from 
each of the centers above described, until the last arc meets the 
circle first drawn. To obtain the centers by which the inner line of 




Fig. 



162 



163. — The Eye of the Volute of Fig. 
Enlarged to Show its Construction. 

of the volute is struck, and which gradually approaches the outer line throughout its course, proceed as follows : 
Produce the line 3 c until it intersects 1 2 in the point 1', which mark. This operation gives also the points 9 1 
and 5' of intersection with the lines parallel to 1 2, which also mark. In like manner produce 4 c, lc and 2 c, 
as shown by the dotted lines, and mark the several points of intersection formed with the cross lines. Then the 
points 1', 2 1 , 3 1 , 4 1 , etc., thus obtained are the centers for the inner line of the volute, which use in the same 
manner as described for producing the outer line. Although this rule is in quite general use for describing 
the scroll sides of brackets and modillions, that given in Section 310 will be found more satisfactory. 



38 



Geometrical Problems 




164. — To Draw a Spiral from 
Centers ivith Compasses. 





308. To Draw a Spiral from Centers with Compasses. — Divide the circumference of the primary — some- 
times called the eye of the spiral — into any number of equal parts ; the larger the number of parts the more 

regular -will be the spiral. Fig. 164 shows the primary divided into six 
equal parts. Fig. 165 is an enlarged view of a portion of the preced- 
ing figure. Complete the polygon by drawing the lines 1 2, 2 3, 3 4, 
etc., producing them outside of the primary, as shown by A, B, D, F, 
C and E. From 2 as center, with 2 1 as radius, describe the arc A B. 
From 3 as center, and 3 B as radius, describe the arc B J) ; and with 4 as 
center, with radius 4 D, describe the arc D F. In this manner the spiral 
may be drawn any number of revolutions. Use 1, 2, 3, 4, 5 and 6 as 
centers, describing from each in turn an arc contained between two sides. 
309. To Draw a Spiral, by Means of a Spool and Thread. — Set the 
spool, as shown by A D 
B in Fig. 166, and wind 
a thread around it. Make 
a loop, E, in the end of 
the thread, in which 
place a pencil, as shown. Hold the spool firmly and move the 
pencil around it, unwinding the thread. A curve will be de- 
scribed, as shown in the dotted lines of the engraving. It is 
evident that the proportions of the figure are determined by 

the size of the spool. Hence 
a larger or smaller spool is to 
be used, as circumstances re- 
quire. 

310. To Draw a Scroll Fig. 
to a Specif ed Width, as for 

a Bracket or Modillion. — In Fig. 167, let it be required to construct a 
scroll which shall touch the line D B at the top, E A at the bottom and 
A B at the side, the length of A B, which determines the length of the top 
and bottom line, being given. Bisect A B, obtaining the point C. Let the 
distance between the beginning and ending of 
the first revolution of the scroll, shown by a e, 
be established at pleasure. Having determined 
this distance, take one-eighth of it and set it off 
upward from C on the line A B, thus obtaining 
the point b. From b draw a horizontal line of any convenient length, as shown 
by b h. "With the point of the compasses set at b, and with b A as radius, de- 
scribe an arc cutting the line b h in the point 1. In like manner, from the same 
center, with radius b B, describe an arc cutting the line b h in 
the point 2. Upon 1 2 as a base erect a square, as shown by 
12 3 4. Then from 1 as center, with 1 a as radius, describe an 
arc a b ; and from 2 as center, with 2 b as radius, describe the 
arc b c. From 3 as center, with radius 3 c, describe the arc 
c d. From 4 as center, with radius 4 d, describe the arc d e. If 

the curve were continued from E, being struck from the same centers, it would run parallel 
to itself ; but as one line of the scroll runs parallel to the outer line, its width may be set 
off at pleasure, as shown by a a 1 , and the inner line may be drawn by the same centers as 
already used for the outer, and continued until it is intersected by the outer curve. To find 
the centers from which to complete the outer curve, construct upon the line of the last 
radius above used (4 e) a smaller square within the larger one, as shown by 5 6 7 8. This 
is better illustrated by the larger diagram, Fig. 168, in which like figures represent the same 
points. Make the distance from 5 to 8 equal to one-half of the space from 4 to 1 ; and make 4 to 8 equal the 



165. — An Enlarged View of the Central Part 
of Fig. 164. 



Fig. 166. — To Draw a Spiral by 
Means of a Spool and Thread. 




Fig. 167. — To Draw a Scroll to 
a Specified Width, as for a 
Bracket or Modillion. 



I 

Fig. 168.— The Cen- 
ter of Fig. 167, 
Enlarged to better 
Illustrate its Con- 
struction. 



Geometrical Problems. 



39 



distance of 5 to 1. Make 5 to 6 equal the distance from 8 to 5. After obtaining the points 5, 6, 7, etc, in this 
manner, so many of them are to be used as are necessary to make the outer curve intersect the inner one as 
shown at g. Tims 5 is used as a center for the arc e f, and 6 as a center for the arc f g. If the distance a a 1 
were taken less than here given, it is easy to see that more of the centers upon the small square would require 
to be used to arrive at the intersection. 

THE CONSTRUCTION OF REGULAR POLYGONS. 



I. BY THE USE OF COMPASSES AND STKAIGHT-EDGE. 

311. The most common rules in use for the construction of polygons, whether drawn within circles or 
erected upon given sides, are those which employ the straight-edge and compasses only. In some instances these 
rules are the best for the pattern cutter to employ. In other cases his ends are better served by rules makin°- 
use of other instruments. Accordingly, we divide our remarks upon the construction of polygons into four 
parts, arranging them according to the tools employed. By this presentment the student will have no difficulty 
in seeing the relative advantages of the different methods, and by becoming expert in the use of different 
instruments, will be able to select the best rules for his purpose as circumstances arise. 

312. To Draw an Equilateral Triangle within a Given Circle. — In Fig. 169, let A B D be any given circle, 

within which an equilateral triangle is to be drawn. From any point in the 
circumference, as E, with a radius equal to the radius of the circle, describe 
the arc D C B, cutting the given circle in the points D and B. Draw the 
line D B, which will be one side of the required -triangle. From D or B as 
center, and with D B as radius, cut the circumference of the given circle, as 
shown at A. Draw A B and A D, which will complete the figure. 




Fig. l6g. — To Draw an Equilateral 
Triangle within a Given Circle. 




Fig. 170. — To Draw a Square 
within a Given Circle. 



313. To Draw a Square within a Given 
Circle.— In Fig. 170, let A C B D be any 
given circle within which it is required to 
draw a square. Draw any two diameters at 
right angles with each other, as C D and A B. 
Join the points C B, B D, D A and A C, 
which will complete the required figure. 

314. To Draw a Regular Pentagon 
within a Given Circle. — In Fig. 171, A D G B C represents a circle in which it 
is required to draw a regular pentagon. Draw any two diameters at right angles 
to each other, as A B and D C. Bisect the radius A H, as shown at E. With 
E D as radius strike the arc D F, and with the chord D F as radius strike 

D the arc F G, cutting the circumference of the 

given circle at the point G. Draw D G, which will equal one side of the 
required figure. With the dividers set equal to D G, step off the spaces in 
the circumference of the circle, as shown 
by the points I K L. Draw D I, I K, 
K L and L G, thus completing the figure. 

315. To Draw a Regular Hexagon with- 
in a Given Circle— In Fig. 172, let A B D 
E F G be any given circle within which a 
hexagon is to be drawn. From any point 
in the circumference of the circle, as at A, 
with a radius equal to the radius of the cir- 
cle, describe the arc C B, cutting the cir- 
cumference of the circle in the point B. 

Then A B will be one side of the hexagon. 




Fig. 171. — To Draw a Regular Pcnta 
gon within a Given Circle. 




Fig. 



172. — To Draw a Regular Hexa- 
gon within a Given Circle. 



Connect the points A and B, 

With the dividers set to the distance A B, step off in the circumference of 

the circle the points G, F, E and D. Draw the connecting lines A G, G F, F E, E D and D B, thus completing 



Geometrical Problems. 



the figure. By inspection of this figure it -will be noticed that the radius of a circle is equal to one side of the 
regular hexagon which may be inscribed within it. Hence it follows that drawing the arc C B may be dispensed 

with. Set the dividers to the radius of a circle and step around the circum- 
ference, connecting the points thus obtained. 

316. To Draw a Regular Heptagon within a Given Circle. — In Fig. 

173, let F A G B H I K L D be the given 

circle. From any point, A, in the cir- 
cumference, with a radius equal to the 

radius of the circle, describe the arc 

BCD, cutting the circumference of the 

circle in the points B and D. Draw the 

chord B D. Bisect the chord B D, as 

ehown at E. "With D as center, and with 

D E as radius, strike the arc E F, cutting 

the circumference in the point F. Draw 

D F, which will be one side of the hep- 
tagon. "With the dividers set to the distance D F, set off in the circumfer- 
ence of the circle the points GHIKL, and draw the connecting lines F G, 

G H, H I, I K, K L and L D, thus com- 
pleting the figure. 

317. To Draw a Regular Octagon within a Given Circle 




Fig. 173. — To Draw a Regular Hepta- 
gon within a Given Circle. 





174. — -To Draiv a Regular Octa- 
gon within a Given Circle. 

In Fu 



Fig. 175. — To Draw a Regular Nona 
gon within a Given Circle. 

within a Given Circle. — In Fig. 175 





Fig. 



174, 
let B I D F A G E H be the given circle within which an octagon is to be 
drawn. Draw any two diameters at right 
angles to each other, as B A and D E. 
Draw the chords D A and A E. Bisect 
D A, as shown, and draw L II. Bisect 
A E and draw K I. Then connect the 
several points in the circumference thus 
obtained by drawing the lines D I, I B, 
B H, II E, E G, G A, A F and F D, 
which will complete the figure. 

318. To Draw a Regular Nonagon 
let M G E be the given circle. Draw 

any two radii at right angles to each other, as B C and A C, and draw the 

chord B A. From A as center, and with 
a radius equal to one-half the chord A B ? 
as shown by A D, strike the arc D E, 

cutting the circumference of the circle at the point E. Draw A E, which 
will be one side of the nonagon. Set the dividers to the distance A E and 
step off the points M, H, K, G, I, F and L, and draw the connecting lines, 
as shown, thus completing the figure. 

319. To Draiu a Regular Decagon within a Given Circle. — In Fig. 176, 

let D B E A be any given circle in which a decagon is to be drawn. Draw 

any two diameters through the circle at right angles to each other, as shown 

by B A and D E. Bisect B C, as shown at F, and draw F D. With F as 

center, and F D as radius, describe the arc D G, cutting B A in the point G. 

Draw the chord D G. With D as center, and D G as radius, strike the arc 

Fig. in— To Draw a Regular Undeca- G H, cutting the circumference in the point H. Connect D and H, as 
gon within a Given Circle. ghown _ B;gect D H and dpaw ^ Hne c ^ cutting the circumference in 

the point I. Draw the lines II I and I D, which will then be two sides of the required figure. Set the divid- 
ers to the distance H I and space off the circumference of the circle, as shown, and draw the connecting lines 
D K, K M, M L, L P, P E, E jST, E" O and H, thus completing the figure. 

320. To Draw a Regular Undecagon within a Given Circle. — In Fig. 177, let B D A L be any given circle 



176. — To Draw a Regular Deca- 
gon within a Given Circle. 



Geometrical Problems. 



41 



as 

as 





Fig. 



__->< 



178. — To Draw a Regular Dodecagon 
within a Given Circle. 



in which a regular figure of eleven sides is to be drawn. Draw any diameter, as B A, and draw a radius. 

D C, at right angles to B A. Bisect C A, thus obtaining the point E. From E as center, and with E D 

radius, describe the arc D F, cutting B A in the point F. With D as 

center, and D F as radius, describe the arc F G, cutting the circumfer- 
ence in the point G. Draw the chord G D and bisect it, as shown by 

H C, thus obtaining the point K. From D as center, and with D K as 

radius, cut the circumference in the point I. Draw I D. Then I D will 

be equal to one side of the required figure. Set the dividers to this 

space and step off the points in the circumference, as shown by N, B, 

S, ~K, B, L, 0, T, J and G, and draw the connecting arcs, as shown, thus 

completing the figure. 

321. To Draw a Regular Dodecagon within a Given Circle. — In Fig. 

ITS, let M F A I be any given circle 
in which a dodecagon is to be drawn. 
From any point in the circumference, 
as A, with a radius equal to the radius 

of the circle, describe the arc C B, cutting the circumference in the point 
B. Draw the chord A B, which bisect as shown, and draw the line O 0, 
cutting the circumference in the point D. Draw A D, winch will then be 
one side of the given figure. With the dividers set to this space step off 
in the circumference the points B, I, N, II, M, G, L, F, K and E, and 
draw the several chords, as shown, thus completing the figure. 

322. General Rule for Drawing any Regidar Polygon in a Circle. — 
Rule. — Through the given circle draw any diameter. At right angles to this 
diameter draw a radius. Divide that radius into four equal parts, and pro- 
long it outside the circle to a distance equal to three of those parts. Divide 
the diameter of the circle into the same number of equal parts as the 
polygon is to have sides. Then from the end of the radius prolonged, as 
above described, through the second division in the diameter, draw a line 
cutting the circumference. Connect this point in the circumference and 
the nearest end of the diameter. The line thus drawn will be one side of 

the required figure. Set the dividers to this space and step off on the circum- 
ference of the circle the remaining number of sides and draw connecting lines, a b 

which will complete the figure. 

323. To Draio a Regular Polygon of Eleven Skies within a Given Circle 
by the General Rule just given. — Through the given circle, E D F G in Fig. 
179, draw any diameter, as E F, which divide into the same number of equal 
parts as the figure is to have sides, as shown by the small figures. At right angles 
to the diameter just drawn draw the radius D K, which divide into four equal 
parts. Frolong the radius D K outside the circle to the extent of three of 
those parts, as shown by a b c, tlras obtaining the point c. From c, through the 
second division in the diameter, draw the line c H, cutting the circumference in 
the point H. Connect H and E. Then H E will be one side of the required 
figure. Set the dividers to the distance II E and step off the circumference, 
as shown, thus obtaining the points for the other sides, and draw the con- 
necting arcs, all as illustrated in the figure. 

324. Ujjon a Given Side to Construct an Equilateral Triangle. — In Fig. 180, let A B represent the length 
of the given side. Draw any line, as C D, making it equal to A B. Take the length A B in the dividers, and 
placing one foot upon the point C, describe the arc E F. Then from D as center, with the same radius, describe 
the arc G.H, intersecting the first arc in the point K. Draw K C and K D. Then C D K will be the required 
triangle. 

325. To Construct a Triangle, the Length of the Three Sides being Given.— In Fig. 181, let A B, C D and 
E F be the given sides from which it is required to construct a triangle. Draw any straight line, G H, making 



Fig. 179. — To Draw a Regular Polygon 
of Eleven Sides within a Given Circle 
by General Rule Given in Section 322. 




Fig. 180. — Upon a Given Side to 
Construct an Equilateral Tri- 
angle. 



42 



Geometrical Problems. 



A 
C 




— B 








U A 








181. — To Construct a Triangle, the 
Length of the Three Sides being Given. 



it in length equal to one of the sides, E F. Take the length of one of the other sides, as A B, in the compasses, 
and from one end of the line just drawn, as G, for center describe an arc, as indicated by L M. Then, setting 

the compasses to the third side, C D, from the opposite end of the line 
first drawn, as H, describe a second arc, as I K, intersecting the first in 
the point 0. Connect O G and O H. Then OGH will be the required 
triangle. 

32C. Upon a Given Side to Draw a Regular Pentagon. — In Fig. 
182, let A B represent the given side upon which a regular pentagon 
is to be constructed. With B as center and B A as radius, draw the 
semicircle A D E. Produce A B to E. Bisect the given side A B, as 
shown at the point F, and erect a perpendicular, as shown by F C. 
Also erect a perpendicular at the jioint B, as shown by G II. With B 
as center, and F B as radius, 
strike the arc F G, cutting 
the perpendicular II G in the point G. Draw G E. With G as cen- 
ter, and G E as radius, strike the arc E II, cutting the perpendicular 
in the point H. With E as center, and E II as radius, strike the arc 
II D, cutting the semicircle A D E in the point D. Draw D B, 
which will be the second side of the pentagon. Bisect D B, as shown, 
at the point K, and erect a perpendicular, which produce until it 
intersects the perpendicular F C, erected upon the center of the given 
side in the point C. Then C is the center of the circle which cir- 
cumscribes the required pentagon. From C as center, and with C B 
as radius, strike the circle, as shown. Set the dividers to the distance 

A B and step off the circumfer- 
ence of the circle, obtaining the 
points M and L. Draw A ~M, 
M L arid L D, which will complete the figure. 

327. Upon a Given Side to Draw a Regular Hexagon. — In Fig. 183, 
let A B be the given side upon which a regular hexagon is to be erected. 
From A as center, and with A B as radius, describe the arc B C. From B as 
center, and with the same radius, describe the arc A C, intersecting the first 
arc in the point C. C will then be the center of the circle which will cir- 
cumscribe the required hexagon. With C as center, and C B as radius, strike 
the circle, as shown. Set the dividers to the space A B and step off the cir- 
cumference, as shown, obtaining the points E, G, F and D. Draw the chords 
A E, E G, G F, F D and D B, thus 
completing the required figure. 
328. Upon a Given Side to Draw a Regular Heptagon.— In Fig. 184, 
A B represents the given side upon which a regular heptagon is to be 
drawn. From B as center, and with B A as radius, strike the semicircle 
A E D. Produce A B to D. From A as center, and with A B as radius, 
strike the arc B F, cutting the semicircle in the point F. Bisect the given 
side A B, obtaining the point G. Draw G F, producing it indefinitely in 
From D as center, and with radius G F, cut the semi- 
. Draw the line E B, which is another side of the 





Fig. 182. — Upon a Given Side to Draw a 
Regular Pentagon. 



Fig. 



183. — Upon a Given Side to Draw 
a Regular Hexagon. 




the direction of C. 
circle in the point E 

required heptagon. Bisect E B, and upon its middle point erect a perpen- 
dicular, which produce until it meets the perpendicular erected upon the 
center of the given side A B in the point C. Then C is the center of the Fij - 
circle which will circumscribe the required heptagon. From C as center, 

and with C B as radius, strike the circle. Set the dividers to the distance A B and step off the circumference 
as shown, obtaining the points K, N, M and L. Draw the connecting arcs A K, K N, N M, M L and L E thus 
completing the figure. J 



1S4. — Upon a Given Side to Draw a 
Regular Heptagon. 



43 

185, let A B represent the given side upon 

From B as 





Fig. 1S6. — Upon a Given Side to Draw a 
Regular Nonagon. 



Geometrical Problems. 

329. Upon a Given Side to Draw a Regular Octagon. — In Fig 
which a regular octagon is to be constructed. Produce A B indefinitely in the direction of D 
center, and with A B as radius, describe the semicircle AED. At the 
point B erect a perpendicular to A B, as shown, cutting the circumference 
of the semicircle in the point E. Bisect the arc E D, obtaining the point 
F. Draw F B, which is another side of the required octagon. Bisect the 
two sides now obtained and erect perpendiculars to their middle points, Gl- 
and H, which produce until they intersect at the point C. C then is the 
center of the circle that will circumscribe the octagon. From C as center, 
and with C B as radius, strike the circle, as shown. Set the dividers to the 

space A B and step off the circumfer- 
ence, obtaining the points L, K, M, 
O and N. Draw the connecting arcs 

A L, L K, Iv M, M 0, IST and N F, Fig. 185.— Dpon a Given Side to Draw a 
thus completing the required figure. Regular Octagon. 

330. Upon a Given Side to Draw a Regular Nonagon. — In Fig. 
186, A B is any given side upon which it is required to draw a regular 
nonagon. Produce A B indefinitely in the direction of D. From B as 
center, and with B A as radius, strike the semicircle A F D. At the 
point B erect a perpendicular to A B, cutting the semicircle in the point 
F. Draw the arc F D, which bisect, obtaining the point G. From D as 
center, and with D G as radius, cut the semicircle in the point E. Draw 
E B, which will be another side of the required figure. From the mid- 
dle points of the two sides now ob- 
tained, as H and K, erect perpen- 
diculars, which produce until they intersect at the point C. Then C is the 
center of the circle which will circumscribe the required nonagon. From 
C as center, and with C B as radius, strike the circle B O P A. Set the 
dividers to the space A B and step off the circle, as shown, obtaining the 
points N, P, M, R, O and L. Draw the connecting chords, A N, 1ST P, 
P M, M R, E 0, O L and L E, thus completing the figure. 

331. Upon a Given Side to Draw a Regular Decagon. — In Fig. 187, 
A B is the given side upon which a regular decagon is to be drawn. Pro- 
duce A B indefinitely in the direction of D. From B as center, and with 
B A as radius, strike the semicircle A H D. Bisect the given side A B, 
obtaining the point F. Through the point B draw the line II B G, per- 
p pendicular to A B. From B as center, 

and with B F as radius, strike the arc 

F G, cutting the perpendicular H G in the point G. From G as center, and 
with G D as radius, strike the arc D O, cutting the perpendicular H G in the 
point 0. From D as center, and with D as radius, strike the arc O K, cut- 
ting the semicircle in the point K. Draw the line K D, which bisect with the 
line B L, cutting the semicircle in the point E. Then E B will be another side 
of the decagon. Upon the middle points, F and M, of the two sides now ob- 
tained erect perpendiculars, which produce until they intersect at the point C. 
Then C is the center of the circle which will circumscribe the required decagon. 
From C as center, and with C B as radius, strike the circle, as shown. Set the 
dividers to the space A B and step off the circle, obtaining the several points, 
I, N, S, V, E, T and P. Draw the connecting lines, A I, I N, N S, S V, Y B 
K T, T P and P E, thus completing the figure. 

332. Upon a Given Side to Draw a Regular Undecagon. — In Fig. 188, 
A B represents the given side upon which a regular undecagon is to be drawn. Produce A B indefinitely in the 
direction of D. From B as center, and with B A as radius, draw the semicircle A M D. Through the point 




Fig. 




187. — Upon a Given Side to Draw 
Regular Decagon. 



Fig. 188. — Upon a Given Side to 
Draw a Regular Undecagon. 



44 



Geometrical Problems. 




-g- a D 

Fig. 189. — Upon a Given Side to 
Draw a Regular Dodecagon. 



B, perpendicular to A B, draw the line H G indefinitely. From B as center, and with B F as radius, strike the 
arc F G, cutting the perpendicular H G in the point G. From G as center, and G D as radius, strike the arc D H, 
cutting the perpendicular H G in the point II. "With D as center, and D H as radius, strike the arc H M, cut- 
ting the semicircle in the point M. Draw M D, which bisect, obtaining the point K, through which, from B, 

draw the line B K, and produce it until it cuts the semicircle in the point E. 
Then B E will be another side of the required figure. Bisect the two sides 
now obtained and erect perpendicular lines, producing them until they inter- 
sect, as shown by F C and L C. Then C, the point of intersection, is the cen- 
ter of the circle which circumscribes the undecagon. From C as center, and 
with C A as radius, strike the circle, as shown. Set the dividers to the space 
A B and step off the circumference, obtaining the points O, V, T, R, P, S, N 
and I. Draw the chords A O, V, V T, TE, B P, P S, S N 3 N I and I E, 
tims completing the figure. 

333. Upon a Given Side to Draw a Regular Dodecagon. — In Fig. 189, 
let A B represent the given side upon which a regular dodecagon is to be 
drawn. Produce A B indefinitely in the direction of D. From B as center, 
and with B A as radius, describe the semicircle A E D. From D as center, and 
with D B as radius, describe the arc B F, cutting the semicircle in the point 
F. Draw F D, which bisect by the line V B, cutting the semicircle in the 
point E. Then E B is another side of the dodecagon. From the middle 
points of the two sides now obtained, as G and H, erect perpendiculars, as 
shown, cutting each other at the point C. This point of intersection, C, then 
is the center of the circle which will circumscribe the required dodecagon. 
From C as center, and with C B as radius, strike the circle, as shown. Set the 
dividers to the distance A B and space off the circumference, thus obtaining 
the points L, P, M, S, 1ST, B, 0, K and I. Draw the connecting lines L P, 
P M, M S, S 1ST, JST R, R O, O K, K I and I E, thus completing the figure. 

334. General Ride by which to Draw any Regular Polygon, the Length 
of a Side being Given. — With a radius equal to the given side describe a semi- 
circle, the circumference of which divide into as many equal parts as the fig- 
ure is to have sides. From the center by which the semicircle was struck 
draw a line to the second division in the circumference. This line will be one 
side of the required figure, and one-half of the diameter of the semicircle 
will be another, and the two will be in proper relationship to each other. 
Therefore, bisect each, and through their centers erect perpendiculars, which 
produce until they intersect. The point of intersection will be the center of 
the circle which will circumscribe the polygon. Draw the circle, and setting 

the dividers to the length of one of the sides already found, step off the 
circumference, thus obtaining points by which to draw the remaining sides 
of the figure. 

335. To Construct a Regular Polygon of Thirteen Sides, the Length of 
a Side being Given, by the General Rule in Section 334. — In Fig. 190, let 
A B be the given side. With B as center, and with B A as radius, describe 
the semicircle A F G. Divide the circumference of the semicircle into thir- 
teen equal parts, as shown by the small figures, 1, 2, 3, 4, etc. From B draw 
a line to the second division in the circumference, as shown by B 2. Then 
A B and B 2 are two of the sides of the required figure, and are in correct 
relationship to each other. Bisect A B and B 2, as shown, and draw D C 
and E C through their central points, prolonging them until they intersect 
at the point C. Then C is the center of the circle which will circumscribe 
the required polygon. Strike the circle, as shown. Set the dividers to the 
space A B, and step off corresponding spaces in the circumference of the circle, as shown, and connect the 
several points so obtained by lines, thus completing the figure. 




Fig. 190. — To Construct a Regular 
Polygon of Thirteen Sides, the 
Length of a Side being Given, by 
the General Rule in Section 334. 




Fig. 191. — Within a Given Square 
to Draw a Regular Octagon. 



Geometrical Problems. 



45 



336. Within a Given Square to Draio a Begular Octagon. — In Fig.'191, let A D B E be any given square, 
within which it is required to draw an octagon. Draw the diagonals D E and A B, intersecting at the point 
C. . From A. D, B and E as centers, and with radius equal to one-half of one of the diagonals, as A C, strike 
the several arcs II N, 6 I, I M and L 0, cutting the sides of the square, as shown. Connect the points thus 
obtained in the sides of the square by drawing the lines 6 O, H I, K L, and M 1ST, thus completing the figure. 

II. — BY THE USE OF THE T-SQUAKE AND TRIANGLES, 0E SET-SQUARES. 

337. In another part of the boob (Section 72) we described the division of the circle, for the measurement 
of angles, into spaces called degrees, and, in connection with our description of drawing tools, we described cer- 
tain triangles or set-squares (Section 233) which are in common use, nam- 
ing them by the degrees which their angles contain. These set-squares, in 
connection with the T-square or a straight-edge, can be used advantageously 
for constructing various polygons, whether inscribed or circumscribed. 
They are derived directly from the circle ; that is, they represent certain 
fixed portions of the circle, and therefore may be employed in dividing a 
circumference for the purpose of constructing polygons. To make their use 
for this purpose entirely clear, we will first describe their origin and after- 
ward give illustrations of their employment. 

33S. Since the circle consists of 360 degrees, a quarter of it is repre- 
sented by 90 degrees. In Fig. 192, 
the circle A C B D is divided into 
quarters by the diameters A B and 
C D, drawn at right angles to each 
other. It will be seen that the same 
result might be accomplished by using 

the set-square A E C, by bringing its right angle to the center of the circle 
E, and scribing along its sides for E and E A, and then shifting it for 
the other parts. The instrument A E corresponds to a quarter circle, and 
is therefore called a 90-degree set-square. If we divide the circle into eight 
equal parts by diameters, as shown in Fig. 193, each angle will represent 
one-eighth of 360 degrees, or 45 degrees. 
Hence, the instrument which corresponds 
to one of these angles, as A E F in Fig. 
193, is called a 45-degree set-square. If 
we divide the circle into twelve equal 
194, each angle will represent one-twelfth 

of the set- 
circle were 





Fig. 192. — A Circle Divided into Four 
Equal Parts by the Use of a qo-Degree 
Set-Square. 



Fig. 193. — .4 Circle Divided into Eight 
Equal Parts by the Use of a ^-De- 
gree Set-Square. 




parts by diameters, as shown in Fi 

of 360 degrees, or 30 degrees, which gives the name to the angle 

square corresponding to it, as shown. In like manner, if the 

divided 'into six equal parts, each of the angles would measure 60 degrees, 

which gives name to another angle of the set-square, which is shown by A B E 

in Fig. 194. Still other set-squares might be employed, but the two which 

contain the four angles we have described are found entirely adequate for all 

ordinary requirements. 

339. A governing principle upon which this use of the set-square depends, 
may be briefly referred to in this connection with advantage. We have de- 
scribed the set-squares as 45, 45 and 90, and 30, 60 and 90 degrees respectively. 
It will be observed that the sums of these sets of figures are the same ; that is, 45 + 45 -4- 90 = ISO, and 30 + 
60 + 90 = 180. Further, it will be discovered, upon investigation, that the sum of the angles of any triangle 
whatsoever also equals 180 degrees. Each of the set-squares contains a right angle. Hence, in working from 
a T-square or other straight line, bv means of it lines may be drawn at right angles, and also at the several 
intermediate angles represented by their other sides. The sum of the angles of the set-squares always being 
ISO degrees, addition, subtraction and division in the calculation of angles become a very simple matter; but 
for the most part these operations are performed graphically, as will appear further on. 



Fig. 194. — A Circle Divided into Twelve 
Equal Parts by the Use of a 30-De- 
gree Set-Square. 



46 



Geometrical Problems. 



340. Inasmuch as eacli of the set-squares contains an angle of 90 degrees, instead of describing them as 45, 
45 and 90-degree, and 30, 60 and 90-degree set-squares, the form is abbreviated in the first instance to a "45-de- 
gree set-square," and in the second to a " 30-degree set-square," or a " 60-degree set-square," as the case may be, 

the latter terms for the second instrument being used interchangeably. With 
a right angle (90 degrees) in the set-square and an angle of 45 degrees, the 
third angle must be 45 degrees also, in order to complete the sum, ISO degrees. 
In like manner, given a set-square with an angle of 90 degrees (a right angle) 
and another of 60 degrees, the remaining angle must be 30 degrees, and vice 
versa. Therefore, no confusion can possibly arise in calling these tools set- 
squares of 45 degrees and of 30 degrees, or 60 degrees, as the case may be. 

341. In describing the angle of 90 degrees in the set-square, we compared 
the division of the circle into four equal parts by two diameters drawn at right 
angles, with the same result acconq:>lished by the use of this tool placed as 
shown in Fig. 192. Such a plan of dividing the circle by the use of the set- 
square, that is, by bringing the right angle of the set-square against its center, 
is quite inconvenient. A better method, and one which makes use of the same 




D F 

Fig. 195. — A Circle Divided into 
Four Equal Farts by a 45-Zte- 
gree Triangle. 




M I N 

Fig. 196. — A Circle Divided into Six Equal 
Parts by the Use of a 30-Degree Set Square. 



principles in the set-square, is shown in Fig. 195. A straight-edge, as, 
for instance, a T-square, is placed tangent to or near the circle, as shown 
by A B. One side of a 45-degree set-square is placed against it, as 
shown, its side C F being brought against the center. The line C F is 
then drawn. By reversing the set-square, as shown by the dotted lines, 
the line E D is drawn at right angles to C F, thus dividing the circle 
into quarters. 

342. A similar use of the second set-square above described is 
shown in Fig. 196, by which a circle is divided into six equal parts. 
Place a straight-edge tangent to or near the circle, as shown by A B. 
Then place the set-square as shown by G B M, bringing the side G B 
against the center of the circle, drawing the line D L. Then place it as shown by the clotted lines, bringing 
tlie side A II against the center, scribing the line F E. Then, by reversing the set-square, placing the side G M 
against the straight-edge, erect the perpendicular C I, completing the division. A few of the problems to which 
these principles may be advantageously applied will now be demonstrated. 

343. To Draw an, Etjuilateral Triangle xoithin a Given Circle. — In Fig. 197, let D be the center of the 

given circle. Set the side C F of 
a 30-degree set-square against the 
T-square, as shown, and move it 
along until the side E G- touches 
D. Mark the point B. Beverse 
the set-square so that the point 
E will come to the right of the 
side F G. Move the set-square 
along in the reversed position un- 
til the side E G again meets the 
point D, and mark the point C. 
Move the T-square upward until 
it touches the point D, and mark 

Figs. 197 and 198. — To Draw an Equilateral Triangle within a Given Circle. +],„ D int A Then A B and C 

are points which divide the circle into three equal parts. The triangle may be easily completed from this 
stage by drawing lines connecting A B, B C and C A, with any straight-edge or rule, but greater accuracy is 
obtained by the further use of the set-square, as follows : Place the side F G of the set-square against the 
T-square, as shown in Fig. 198, and move it along until the side E G touches the points A and C, as shown. 
Draw A C, which will be one side of the required triangle. Set the side E F of the set-square against the 
T-square, and move it along until the side F G coincides with the points C and B. Then draw C B, which will 
be the second side of the triangle. Place the side F G of the set-square against the T-square, with the side 




Geometrical Problems. 



47 



E F to the right, and move it along until the side E G coincides with the points A and B. Then draw A B 
thus completing the figure. The same results may be accomplished by first establishing the point A, by bring! 
bag the T-square against the center and using the set-square, as shown in Fig. 198. "We present the different 
methods here given, in order to more clearly illustrate the use of the tools employed. 

344. To Draw a Square within a Given Circle.— Let D, in Fig. 199, be the center of the given circle. 
Place the side E F of a 45-degree 
set-square against the T-square, as 
shown, and move it along until 
the side E G meets the point D. 
Mark the points A and B. Be- 
verse the set-square, and in a sim- 
ilar manner mark the points C 
andH. The points A, H, Band C 
are corners of the required square. 
Move -the T-square upward until 
it coincides with the points A 
and II and draw A H, as shown 
in Fig. 200. In like manner 

draw C B. "With the side E F Figs - J 99 and 200.— To Draw a Square within a Given Circle. 

of the set-square against the T-square, move it along until the side G F coincides with the points B and II, and 
draw B II. In a similar manner draw C A, thus completing the figure. 

345. To Draw a Hexagon within a Given Circle. — In Fig. 201, let O be the center of the given circle. 

Place the side E F of a 30- 
degree set-square against 





the Tsquare, 



shown. 



Move the set-square along 
until the side E G meets 
the point O. Mark the 
points A and B. Beverse 
the set-square, and in like 
manner mark the points C 
and D. With the side F G 
of the set-square against the 
T-square, move it along un- 
til the side E F meets the 

Fiqs. 201 and 202. — To Draw a Hexagon within a Given Circle. • ± r\ i it i tt 

" point O, and mark I and Ii. 

Then A, H, D, B, I and C represent the angles of the proposed hexagon. From this stage the figure may be 
readily finished by drawing the sides by means of these points, using a simple straight-edge ; but greater accu- 
racy is attained in completing the figure by the further use of the set- 
square, as shown in Fig. 202. "With the side E F of the set-square 
against the T-square, as shown, draw the line II D, and, by moving the 
T-square upward, draw the side C I. Reversing the set-square so that the 
point G is to the left of the point E, draw the side A II, and also, by 
shifting the T-square, the side I B. With the edge E F of the set-square 
against the T-square, move it up until the side G F coincides with the 
points B and D, and draw the side B D. In like manner draw A C, thus 
completing the figure. In this figure, as with the triangle, the same 
results may be reached by establishing some point, as II, by means of a 
diameter drawn at right angles to the T-square, as shown in the engrav- 
ings, and using it as a base, employing the set-square, as shown in 
Fig. 202. The combination method we have shown is, however, to be 
preferred in many instances, on account of its greater accuracy. Fig. 203.— To Draw an Octagon within a 

346. To Draw an Octagon within a Given Circle.— la Fig. 203, let Gixen Circle - 

K be the center of the given circle. Place a 45-degree set-square as shown in the engraving, bringing its long 




43 



Geometrical Problems. 



side in contact -with the center, and mark the points E and A. Keeping it in the same position, move it along 

until its vertical side is in contact with K, and mark the points D and IT. Reverse the set-square from the 

position shown in the engraving, and mark the points C and G. Move 
the J-square upward until it touches the point K, and mark the points 
B and F. Then A, II, G, F, E, D, C and B are corners of the octa- 
gon. The figure may now he readily completed by drawing the sides, 
by means of these points, using any rule or straight-edge for the pur- 
pose, all as shown by A H, H G, G F, F E, E D, D G C B and B A. 
347. To Draw an Equilateral Triangle upon a Given Side. — In 
Fig. 204, let A B be the given 
side. Set the edge C B of a 
30-degree set-square against 
the T-square, and move it along 
until the edge B D meets the 
point B, and draw the line 
B F. Reverse the set-square, 
still keeping the side C B 
against the T-square, and move 
it along until the side B D 

meets the point A, and draw the line A F, thus' completing the figure. 
348. To Draw a Square upon a Given Side. — In Fig. 205, let 

A B be the given side. Set the edge E F of a 45-degree set-square 

against the T-square, as shown, and move it along until the side E G 

meets the point B, and draw B I indefinitely. Reverse the set-square, 

and bringing the side E G against the point A, draw A F indefinitely. 




Fig. 204. — To Draw an Equilateral Triangl, 
upon a Given Side. 




Fig. 205. — To Draw a Square upon a 
Given Side. 




Fig. 206. — To Draw a Hexagon upon a 
Given Side. 



Bring the T-square against the point B and draw B F. producing it 
until it meets the line A F in the point F. In like manner draw 
A I, meeting the line B G in the point I. Then with the set-square, 
placed as shown in the engraving, connect I and F, thus completing 
the required figure. 

349. To Draw a Hexagon upon a Given Side. — In Fig. 206, 
let A B be the given side. 
Set the edge G H of a 30- 
degree set-square against the 
T-square, as shown, and move 
it along until the edge I G 
coincides with the point A, 
and draw the line A D indefi- 
nitely. Reverse the set-square, 
still keeping the edge G H 
against the T-square, and move it along until the side I G coincides with 
the point B, and draw B E indefinitely. These lines will intersect in the 
point 0, which will be the center of the required figure. Still keeping 
the edge G H of the set-square against the f-square, move it along until 
the perpendicular edge I II meets the point 0, and through O draw F C 
indefinitely. Slide the set-square along until the edge I G meets the 
point B, and draw B C, producing it until it meets the line F C in the 
point C. Reverse the set-square, still keeping the edge G H against the T-square, and draw the line C D, pro- 
ducing it until it meets the line A D in the point D. Slide the set-square along until the side I II meets the 
point D, and draw the line D E, meeting the line B E in the point E. Move the set-square along until the edge 
I G meets the point E, and draw the line E F, meeting the line C F in the point F. Reverse the set-square and 
slide it along until the edge F G meets the point F, and draw F A, meeting the given side in the point A, thus 
completing the required figure. 

350. To Draw cm Octagon upon a Given Side.— In Fig. 207, let C D be the given side. Place one of the 





''•■-.- -- 
















G 

H 

\ 


V 


C, 

J 


A 






/„--'- 










- - 1 


— 






1 



Fig. 207.- 



■To Draw an Octagon upon a 
Given Side. 



Geometrical Problems. 



49 




Figs. 208 and 209.— To Draw an Equilateral Triangle about a Given Circle. 



short sides of a 45-clegree set-square against the T-square, as shown in the engraving. Move the set-square along 
until its long side coincides with the point C. Draw the line C B, and make it in length equal to C D. "With 
the T-square draw the line A B, also iu length equal to C D. Beverse the set-square, and bring the edge ao-ainst 
the point A. Draw A IE in length the same as C D. Still keeping a short side of the set-square against the 
T-square, slide it along until the other short side meets the point H, and draw H G, also of the same length. 
Then, using the long side of the set-square, draw G F of corresponding length. By means of the T-square 
draw F E, and by reversing the set-square draw E D, both in length equal to the original side, C D, joining it 
in the point D, thus completing the required octagon. 

351. To Draw an Equilateral Triangle about a Given Circle. — In Fig. 208, let be the center of the 
given circle. Flace the edge 

E F of a 30-degree set-square 
against the T-square, as shown, 
and move it along until the 
edge F G meets the center O, 
and mark the point A. Be- 
verse the set-square, still keep- 
ing the edge E F against the 
T-square, and in like manner 
mark the point B. Move the 
T-square upward until it meets 
the point O, and mark the 
point C. The required figure 
will be described by drawing 
lines tangent to the circle at 
the points A, B and C, which may be done iu the manner following, as indicated in Fig. 209. Flace the edge 
E G of the set-square against the T-square, and slide it along until the edge F G touches the circle in the point 
B. Draw I Iv indefinitely. Beverse the set-square, keeping the same edge against the T square, and move it 
along until its edge F G touches the circle in the point A, and draw I L, intersecting I K in the point I, the 
other end being indefinite. Then, placing the edge F E of the set-square against the T-square, bring its edge 
E G against the circle in the point C, and draw L Iv, intersecting I L in the point L and I K in the point K, 
thus completing the figure. 

352. To Draw a Hexagon about a Given Circle. — In Fig. 210, let be the center of the given circle. 

Flace the edge E F of a 30- 
degree set-square against 
the T-square, and slide it 
along until the edge F G 
meets the point O, and mark 
the points B and A. Be- 
verse the set-square, still 
keeping the edge E F 
against the T square, and 
in like manner mark the 
points C and D. Bring the 
edge of the T-square against 
O, and mark the points I 
and K. Then C, A, K, D, 
B and I are six points in 

the circumference of the circle, corresponding to the six sides of the required figure. The hexagon is com- 
pleted by drawing a side tangent to the circle at each of these several points, which may be done by using the 
set-square as follows, and as shown in Fig. 211: With the edge E G of the set-square against the T-square, 
bring the edge F G against the circle at the point C, as shown, and draw L M indefinitely. Beverse the set- 
square, and in like manner bring it against the circle at the point A, and draw M X, cutting L M in the point 
M, and extending indefinitely in the direction of 1ST. Slide the set-square along until the edge E F meets the 




Figs. 210 and 211. — To Draw a Hexagon about a Given Circle. 



50 



Geometrical Problems. 



circle in the point K, and draw N" P, intersecting M N in the point 1ST, and extending in the direction of P 
indefinitely. Still keeping the edge E G of the set-square against the T-square, slide it along until the edge 
F G meets the circle in the point D, and draw P P, cutting N P in the point P, but being indefinite in the 
direction of P. Peverse the set-square, and in like manner draw P S tangent to the circle in the point B, cut- 
ting P P in the point R, and extending in the direction of S indefinitely. Slide the set-square along, until its 
edge E F meets the circle in the point I, and draw S L, cutting P S in the point S and L M in the point L, 
thus completing the required figure. 

353. To Dram an Octagon about a Given Circle. — In Fig. 212, let be the center of the given circle. 

"With the edge E F of a 45-de- 
gree set-square against the 
T-square, as shown, move it 
along until the side E G meets 
the point O, and mark the 
points A and P. Peverse the 
set-square, and in like manner 
mark the points C and D. 
Slide the set-square along un- 
til the vertical side G F meets 
the point 0, and mark the 
points H and I. Move the 
T-square up until it meets the 
point 0, and mark the points 
K and L. Then A, I, D, L, 
B, H, C and K are points in 





Figs. 212 and 213. — To Draw an Octagon about a Given Circle. 



the circumference of the given circle corresponding to the sides of the required figure. The octagon is then 
to be completed by drawing lines tangent to the circle at these several points, as shown in Fig. 213, which may 
be done by the use of the set-square, as follows : "With the edge E F of the set-square against the T-square, as 
shown, bring the edge E G against the circle in the point D, and draw M ~N indefinitely. Sliding the set-square 
along until the vertical edge F G meets the circle in the point L, draw N P, cutting M N in the point N, and 
extending in the opposite direction indefinitely. Peverse the set-square, and bringing the edge E G against the 
circle in the point P, draw P P, cutting N" P in the point P, and extending indefinitely in the direction of P. 
Move the T-square upward until it meets the circle in the point H, and draw the line S P, meeting P P in the 
point P, and extending indefinitely in the opposite direction. Then, 
with the set-square placed as shown in the engraving, move it until its 
edge E G meets the circle in the point C, and draw S T, meeting S P 
in the point S, and continuing indefinitely in the direction of T. With 
the set-square in the same position, move it along until its edge G F 
meets the circle in the point Iv, and draw T U, cutting S T in the point 
T, and extending in the opposite direction indefinitely. Peverse the 
set-square, and bringing its long side against the circle in the point A, 
draw U V, cutting T U in the point TJ, and continuing indefinitely 
in the opposite direction. Bring the T-square against the circle in the 
point I, and draw V M, connecting TJ V and M 1ST in the points Y and 
M respectively, thus completing the figure. The above ride will be 
found very convenient for use, although, as the student may discover, 
some points are obtained in the first operation not absolutely necessary. 
354. To Draw a Square about a Gvven Circle. — In Fig. 214, let 
O be the center of the given circle. Place the blade of the T-square against the point 0, and draw the line 
A B. With one of the shorter sides, E F, of a 45-degree set-square against the T-square, and with the other 
short side against the point O, draw the line DOC. Move the T-square upward until it strikes the point C, and 
draw the line II C I. Move it down until it strikes the point D, and draw the line EDK "With the side E F 
of the set-square against the T-square, as shown in the engraving, bring the side E G againt the point A, and 
draw E A II. In like manner bring it against the point B, and draw K B I, thus completing the figure. It is 




Fig. 214. — To Draw a Square about a 
Given Circle. 



Geometrical Problems. 



51 



to be observed that the several lines composing the sides of the square are tangent to the circle in the points 
ACBD respectively. The only object served by drawing the diameters A B and C D is that of obtaining 
greater accuracy, in locating the points just named, than it is possible to secure in drawing the figure around the 
circle without them. 

355. To Draw a Square upon a Given Side. — Let A B of Fig. 215 be the given side. Place one of the 
shorter edges of a 45-degree set-square against the T-square, as placed for 
drawing the given side, and slide it along until the long edge touches the 
point A, and draw the diagonal line A C indefinitely. Place the T-square 
so that its stock comes against the left side of the board, as shown by the 
dotted lines in the engraving, and, bringing the blade against the point A, 
draw A D indefinitely. Then bringing the blade against the point B, draw 
B C, stopping this line at the point of intersection with the line A C, as 
shown at C. Bring the T-square back to the original position and draw the 
line C D, thus completing the figure. In the case of a large drawing board, 
unless the figure is to be located very near one corner of it, and in the case 
of a drawing board of which the adjacent sides are not at right angles, it 
will be desirable to use the right angle of the set-square, instead of changing 
the T-square from one side to the other, as above described. The object of 
drawing the diagonal line A C is to determine the length of the side C B. 
This also may be done by the use of the compasses instead of the set-square, as shown by the dotted arc A 0. 
Prom B as center, with B A as radius, describe the arc A O C. Place the T-square as shown by the clotted lines, 
and, bringing it against the point B, draw B C, producing it until it intercepts the arc A O C in the point C. 
The remaining steps are then to be taken in the manner above described. 




Fig. 215. — To Draw a Square iqjon 
a G-iven Side. 



III. — BY MEANS OF THE PEOTEACTOR. 



356. The protractor, which has been already described (Section 241), is an instrument for measuring 
angles. The most usual form in which this instrument is constructed is that of a semicircle with a graduated 
edge, the divisions being more or less numerous, according to the size of the article. In instruments of ordinary 
size the divisions are single degrees, numbered by 5s or by 10s, while in larger sizes the divisions are made to 
fractions of degrees. 

357. Since the protractor by its divisions represents the divisions of the circle, it may be conveniently em- 
ployed in the construction of polygons. It is especially useful in drawing polygons within given circles, but it 
may also be employed in drawing polygons about given circles, as well as for constructing them upon given 
sides. The latter two cases we shall not attempt to illustrate, as they are rules less advantageous for the pattern 
cutter's use than other methods of doing the same thing elsewhere described in this work. Of the first, namely, 
constructing polygons within given circles, we shall give a few instances, enough to illustrate the use of the 
instrument in a maimer which will enable the reader to make application in other cases as they may arise. 

358. The general plan of using the protractor may be described as measuring from a given point, which 
?*epresents one angle of the required figure, by means of the degrees marked upon it, to another point, and so 
on until the circuit of the circle is completed. Thus, in an equilateral triangle, three spaces of 120 degrees are 
required (3 X 120 — 360), and in a square, four spaces of 90 degrees are required (1 X 90 = 360), while in an 
octagon, eight spaces of 45 degrees are required (8 X 45 = 360), and so on for other polygons. 

359. Since for the purposes of pattern cutting, and perhaps also in some other instances, it is desirable to 
have one side of the polygon fall either to the right or to the left of the figure and parallel to a vertical line 
drawn through the center of it, there are some points to be observed in the manner of making application 
of the simple principles just described, which we will attempt to make plain in the few demonstrations 
following. 

360. To Draw an Equilateral Triangle within a Given Circle. — In Fig. 216, let O be the center of the 
given circle. Through O draw a diameter, as shown by C O D. Place the protractor so that its center point 
shall coincide with O, and turn it until the point marking 60 degrees falls upon the line COD. Then mark 
points in the circumference of the circle corresponding to O and 120 degrees of the protractor, as shown by 
B and E respectively. Draw the lines C E, E B and B C, thus completing the required figure. The reasons 



52 



Geometrical Problems. 



for these several steps are quite evident. The circle consists of 360 degrees. Then each side of an equilateral 
triangle must represent one-third of 360 degrees, or 120 degrees. "We assume the point C for one of the angles, 

and draw the line COD. Then, by the nature of the figure to be 
drawn, D must fall opposite the center of one side. Therefore, 
since 60 is the half of 120 (the length of one side in degrees) we 
place 60 opposite the point D, and mark and 120 for the other 
angles. We then complete the figure by drawing the lines as shown. 
Since in many cases the protractor is much smaller than the circle 
in which the figure is to be constructed, it becomes necessary to 
mark the points at the edge of the instrument, and carry them to 
the circumference by drawing lines from the center of the circle 
through the points, producing them until the circle is reached. 

361. To Draw a Square within a Given Circle. — In Fig. 217, 
let O be the center of the given circle. Through O draw a diam- 
eter, as shown by C O D. Place the protractor so that its center 
point coincides with O, and turn it until the point marking 45 de- 
grees falls upon the line COD.- Mark points in the circumference 
of the circle corresponding to 0, 90 and 180 degrees of the protrac- 
From G, through the cen- 




Fig. 216.- 



-To Draw on Equilateral Triangle 
within a Given Circle. 




Fig. 217. — To Draw a Square within a 
Given Circle. 



tor, as shown by F, G and E respectively. 

ter O, draw G O H, cutting the circumference of the circle in the 
point II. Then E, G, F and II are the angles of the required figure, 
which is to be completed by drawing the sides E G, G F, F Ii and 
H E. Since the circle is composed of 360 degrees, one side of an 
inscribed square must represent one-fourth part of 360 degrees, or 90 
degrees. The half of 90 degrees is 45 degrees. Hence, in setting the 
protractor we placed the point representing 45 degrees opposite the 
point in which we desired the center of one of the sides to fall, or, in 
other words, upon the line COD., Then, having marked points 90 
degrees removed from each other, or, as explained above, opposite the 
points 0, 90 and 180 of the protractor, as shown by F, G and E, the 
fourth point was obtained by the diagonal line. It is evident that H 
must fall opposite G, upon a line drawn through the center. Or we 
might have accomplished the same by moving the protractor around, 
and by means of it measured a space of 90 degrees from either F or 

E, which, as will be clearly seen, would have given the same point, H. 
362. To Draw an Octagon within a Given Circle. — Through the 

center O of the given circle, Fig. 21 S, draw a diameter, AOB, npon 
which the center of one side is required to fall. Place the protractor so 
that its center point shall coincide with the center O, and turn it so that 
the point representing 22^ degrees shall fall on the line AOB. Then 
mark points in the circumference of the circle corresponding to 0, 45, 
90, 135 and 180 degrees of the protractor, as shown by E, G, H, I and 

F. Peverse the protractor, and in like manner mark the points M, L 
and K ; or these points may be obtained by drawing lines from I, H and 
G respectively through the center O, cutting the circumference in M, L 
and K. The figure is to be completed by drawing the sides F I, I H, 
H G, G E, E H, M L, L K and K F. Since the circle consists of 360 
degrees, an octagon must represent 45 degrees, or one-eighth of 360, in 
each of its sides. The half of 45 is 22^-. Hence, we placed the point 
of the protractor representing 22J degrees upon the line AOB, which 

represents the center of one side of the required figure. Having thus established the position of one side, the 
other sides of the figure are located by marking points in the circumference of the circle opposite points in the 
protractor at regular intervals of 45 degrees. 




Fig. 218. — To Draw an Octagon within a 
Given Circle. 



Geometrical Problems. 



53 



363. To Draio a Dodecagon within a Given Circle.— In Fig. 219, let be the center of the given circle. 
Through draw the Diameter A B, at right angles to which one of the sides of the polygon is required to 
be. Set the protractor so that the center point of it coincides with 
the center 0, and revolve it until the point marking 15 degrees falls 
upon the line A O B. With the protractor in this position, mark 
points in the circumference of the circle opposite the points in the 
protractor representing 0, 30, 60, 90, 120, 150 and 180 degrees, as 
shown by E, F, G, II, I, K and L. Then these points will repre- 
sent angles of the required polygon. The remaining angles may 
be obtained by placing the protractor in like position in the oppo- 
site half of the semicircle, or they may be determined by drawing- 
lines from the points F, G, H, I and K through the center O, pro- 
ducing them until they cut the circumference in the points M, 1ST, 
P, Pw and S, which are the remaining angles. The figure is now to 
be completed by drawing the sides, as shown. In a dodecagon, or 
twelve-sided figure, each side must occupy a space represented by 
one-twelfth of 360 degrees, or 30 degrees of the protractor. As 
the side F E was required to be located in equal parts upon opposite 
sides of A O B, we placed the middle of one division of the protractor representing a side (that is, 15 decrees, 
or one-half of 30 degrees) upon the line A O B. Having thus established the position of one side, the others 
are measured off in the manner above described. 




Fig. 2ig.- 



P P. 

-To Draw a Dodecagon within a 
Given Circle. 



TV. BY THE USE OF THE CARPENTER^ SQUARE. 

364. All of the regular polygons may be constructed by the use of a carpenter's square, and the employ- 
ment of this tool for the purpose is frequently of great advantage to the pattern cutter. We shall not attempt 
to give rules for all of the polygons which occur in regular work, but shall limit our remarks, presenting only 
so much as is necessary to illustrate the principles upon which the use of this tool depends. We append a table 
showing the figures upon the square to be used for some of the other polygons than those we describe in full, thus 
enabling any one who is so disposed to experiment further than here illustrated. 

365. To Construct an Equilateral Triangle, the Length of a Side leing Given. — In Fie;. 220, let A B 

in the straight line D C be the length of 
the given side. With 12 of the blade 
placed against the line D C, and with 7 of 
the tongue brought against the point B, 
draw the line B E indefinitely. Reverse 
the square, as shown by the dotted lines, 
maintaining the same points, but bringing 
7 of the tongue against the point A, and 
draw A E, which produce until it cuts the 
line B E, previously drawn in the point E. 
Then A E B will be the required equilateral 
triangle. 

366. To Construct a Hexagon, the 
Length of a Side leing Given. — In Fig. 
221, let B E in the line G II be the length 
of the given side. Take 12 on the blade of the square and 7 of the tongue, and placing the latter against the 
point D, bring the former to the line G II, as shown in the engraving. Then draw the line D C, making it in 
length equal to the given side. Next place the square, as shown by the dotted lines, with 12 of the blade 
against the line G II and 7 of the tongue against the point E, and draw E F, which also make equal to the 
given side. Continue in this way until the several sides of the figure are drawn. In pattern cutting the me- 
chanic more frequently requires the joint line than the outline of the figure itself. The use of the square affords 
him a ready means of obtaining this, without the tedious process of first laying off the polygon. In the case 




Fig. 220. — To Construct an Equilateral Triangle, the Length of a Side 

being Given. 



54 



Geometrical Problems. 



of the figure we have just described, since a hexagon is composed of six equilateral triangles, it follows that 
what we have shown in Fig. 220 is all that is necessary when the miter joints in this shape are required. We 

will, however, for the sake of better 
illustrating this principle, introduce 
an additional engraving, showing a 
different mode of constructing a hex- 
agon from that just described. 

367. To Construct a Hexagon by 
Means of Six Equilateral Triangles, 
the Length of a Sicle being Given. — 
In Fig. 222, let C D in the line A B 
be the length of the given side. Place 
7 of the tongue and 12 of the blade 
against the line A B, as shown, mak- 
ing the latter point fall upon -C, and 
draw C 1ST indefinitely. Next place 
the square, as shown by the dotted 

Draw 




Fig. 221.— To Construct a Hexagon, the Length of a Side being Given. 
lines, with 12 of the tongue and 7 of the blade against the line A B, the latter point falling upon D 



D R indefinitely, cutting the line C N, previously drawn, in the point E. Then E is the center of the circle 
which, if drawn, will circumscribe the hexagon. From E as center, with E C as radius, draw the circle 
C F II K G D. Take the length C D in the dividers, and step off the circle for the other points. It is evident, 
upon inspection, that by producing the line C JST and D R until they cut opposite sides of the circle, the 
points II and X will be obtained, thus 
making it necessary to determine only 
the points F and G by means of the 
dividers. From what has preceded, it 
is also evident that it makes no differ- 
ence upon which arm of the square 
the longer dimension is taken. The 
principle involved is simply that of a 
right-angled triangle and its hypoth- 
enuse. Other lengths than those we 
have described may be employed for 
the purposes indicated, it being neces- 
sary simply to maintain like propor- 
tions. In the above problems we have 
used 12 on one arm of the square, 
suiting the length on the other to it. 
In the table given below we have 
also pursued the same plan. We ad- 
vise the use of 12 as one of the di- 
mensions, because it is easily kept in 
mind, and therefore somewhat sim- 
plifies the rules. 

36S. The following table shows 
the divisions upon the square to be 




Fig. 222. — To Construct a Hexagon by Means oj 
Six Equilateral Triangles, the Length of a 
Side being Given. 



used for constructing some of the polygons which are of very frequent occurrence in pattern cutting. 
Five sides, pentagon, for the figure use 12 on one arm and 3-J on the other. 



Seven 

Eight 

Five 

Seven 

Eight 



Q 9 



heptagon, « " " 12 " " 

octagon, " " " 12 " " 12 " 

pentagon, for the joint line use 12 on one arm and Sf on the other. 

heptagon, " " " 12 " " 5| 



octagon, 



« 



a 
u 



12 






Geometrical Problems. 



55 




iii f 




Fir;. 223. — The Polygon in Position, 
as Drawn by Some of the Rules. 



Mathematical accuracy is not claimed for these rules, although they approach the correct result so closely that 
with ordinary measuring appliances the difference can scarcely be detected. They are sufficiently accurate for 
all the purposes in connection with sheet-metal pattern cutting. 

369. Adjusting the Drawing of a, Polygon to Suit the Requirements of Miter Cutting. — By rules com- 
monly employed for drawing polygons, the figures are frequently so turned 
as to prevent the use of a T-square from the sides of the board for droppiug 
points, and drawing the stretchout and measuring lines. The plans are pro- 
duced as shown in Fig. 223, while for convenience in pattern cutting they 
should be as shown in Figs. 224 and 225. 

370. There are two ways of overcoming this difficulty. One is by 
redrawing the figure, and the other by shifting the paper. The former, 

while it involves considerably more work 
than the latter, is more frequently em- 
ployed than it, because of other draw- 
ings or lines upon the same sheet, as, for 
instance, the elevation or profile, which 
would not be in correct position for \ise 
after the paper was shifted. 

371. To redraw the figure, proceed as follows : Take the length of one 
side in the dividers, as E F, Fig. 223, and bringing the T-square across the 
edge of the circle, as shown in Fig. 224, move it until it cuts the circumfer- 
ence in a chord, the length of which is equal to E F. Draw the line E F, 
which will be one side of the required 
figure in correct position. Then step off 
the circle in the usual manner for the 
other sides. For miter cutting, the side E F and the two radii are alone suf- 
ficient, as will be explained further on. 

372. It may be observed in this connection that when a polygonal plan 
is being drawn for the purpose of miter cutting, and which it is known will 
not be in the proper position when finished, it is not necessary to proceed 
with the figure further than to obtain the length of one side and the radius 
of the circumscribing circle, before making the adjustment by means of 
the T-square, as illustrated in Fig. 224. 

373. To bring the figure into proper position by shifting the paper, 
which is illustrated in Fig. 225, proceed as follows: Flace the T-square in 
position against one side of the board. Then bring one side of the polygon 
against the edge of the blade of the T-square, as shown by D E in the 
engraving. Carefully hold the paper in this position while fastening it at the corners with thumb tacks. 



Fig. 224. — Putting the Plan in Cor- 
rect Position by Redrawing. 




Fig. 225. — Shifting the Position by 
Moving the Sheet of Paper. 



THE ELLIPSE. 

374. Perhaps we cannot do better, in explaining this figure, which in one form or another is so common in 
the pattern cutter's work, than to put our remarks in the shape of a familiar talk about it, giving illustrations 
first of the definitions of an ellipse, and following with several of the methods in common use for drawing it. 

375. A definition of the ellipse very frequently encountered is, "a figure bounded by a regular curve, gen- 
erated from two points, called foci." The idea presented to the mind by this definition immediately contrasts 
the ellipse with the circle. Both are figures bounded by regular curves, but while the ellipse is generated from 
two points, the circle is generated from only one. To carry this comparison a step further, in order to make the 
properties of the ellipse more apparent than perhaps we can do in any other way, let us consider for a moment 
how a circle is drawn by the use of a string and pencil, and then we will see how an ellipse may be drawn by 

the same means. 

376. To draw a circle by a string and pencil, we first determine where we want the center of the figure, 
and then, fastening one end of the string at that place, we attach the pencil to the string at a point just as far 



56 



Geometrical Problems. 




Fig. 226. — To Draw an Ellipse to Specified 
Dimensions with a String and Pencil. 



removed from the center as one-half of the diameter of the circle we propose to draw. After the pencil and 
string are thus arranged, we move the pencil around the center, keeping the string straight all the time. Or, 
to state it in a little different form, if we desire to draw a circle twelve inches in diameter, we tie the pencil to 
the string six inches removed from the center, and then, while keeping the string taut, move the pencil. The 
resulting line will be a circle. 

377. In Fig. 226 is shown the method of drawing an ellipse with string and pencil. By examination of 

the engraving it will be seen that the string (represented by the dotted 
lines) is controlled by the two points F and G, which, as already stated, 
are called foci. To arrange these points, and to adjust the string so as 
to produce a figure of specified dimensions, constitutes the art of draw- 
ing an ellipse with string and pencil. In drawing a circle by the plan 
described, there being but one point and but one dimension, the calcu- 
lations required in getting the position of the pencil with relation to 
the center are very simple. In drawing an ellipse by the same general 
method, there being two points which, by means of the string, control 
the pencil, and two dimensions to the figure to be produced, the calcu- 
lations are a little more complex. Having thus indicated some of the 
points of similarity and contrast between the circle and the ellipse, we 
think the following rule for drawing an ellipse with string and pencil 
will be readily comprehended. 

378. To Draw an Ellipse to Specified Dimensions with a String and Pencil. — In Fig. 226, let it be 
required to draw an ellipse, the length of which shall be equal to the line A B, and the width of which shall be 
equal to the line D C. Lay off A B and D C at right angles to each other, their middle points intersecting, as 
shown at E. With the compasses set to one-half the length of the required figure, as A E, and from either D 
or C as center, strike an arc, cutting A B in the points F and G. These points, F and G, then are the two foci, 
into which drive pins, as shown. Drive a third pin at C. Then pass the string around the three points F, G 
and C and tie it, Eemove the pin C and substitute the pencil, as shown by P. 

379. Another definition of the ellipse, which Fig. 226 also illustrates, and which we call attention to at 
this time because it explains the reason for some of the steps we have just described, is, "a figure bounded by 
a regular curve, from any point in which, if straight lines be drawn to two fixed points, their sum will always 
be the same." A moment's examination of the engraving will demonstrate this. If we take the dividers and 
set off on a straight line the lengths from F and G to the several jioints in the boundary of the figure which 
can be conveniently measured, we shall find their sums equal. For example, the sums of P F and P G, A F 
and A G, C F and C G, B F and B G, are all the same. 

380. ISTow, without stopping to demonstrate it, we will simply call attention to a fact, which is quite evi- 
dent upon inspection of the engraving, and which can be readily proven by 
the use of the dividers. The sum of the distance from any point in the 
boundary to the two foci is equal to the length of the figure. In other 
words, the sum of P F and P G, or C F and C G, is equal to the length 
A B. By inspection of the figure it is evident that each of the two foci 
must be equally distant from the extreme point in the side of the figure, as, 
for instance, C. Therefore the distance from C to F and from C to G must 
each be equal to half of the length of the figure. Hence, in order to 
obtain the position of F and G, we take one-half of A B in the dividers, 
and, with C as center, cut A B by the arc in these points. 

381. An ellipse is sometimes described as " a figure bounded by a regu- 
lar curve, generated from a moving center." This definition necessarily 
implies that the movement of the center and of the point or pencil which describes the curve must be entirely 
in harmony with each other. The most convenient illustration of this definition which can be given, is a 
description of the use of a trammel for drawing an ellipse to given dimensions. 

382. In Fig. 227 we show a trammel as commonly constructed. E is a section through the armSj showing 
the groove in which the head of the bolt F moves. II and G are bolts and pins by which the movement is 
controlled and regulated. In the eneravins: the bar K is shown with holes at fixed distances, through which 




D 



EM <£* 
E F 

Fig. 227. — To Draw an Ellipse to 
Given Dimensions by Means of a 
Trammel. 



Geometrical Problems. 



57 



the governing pins are passed. An improvement upon this plan of construction consists of such a device in 
connection with the pins as will clamp them firmly to the bar at any point, thus providing for an adjustment of 
the most minute variations. 

3S3. Preferring now to the definition of the ellipse before given, II may be regarded as the moving center 
from which the curve bounding the figure is generated. Its motion is lengthwise of the figure, or, in other 
words, from A toward B while describing the upper part of the curve, and the reverse while describing the 
lower part. G is simply the regulator or governor by which harmony of movement is maintained between the 
center II and the pencil I. We will now give the rule for drawing an ellipse with a trammel. 

384. To Draw an Ellipse to Given Dimensions by Means of a Trammel.— In Fig. 227, let it be required 
to describe an ellipse, the length of which shall be equal to A B and the breadth of which shall be C D. Draw 
A B and C D at right angles, intersecting at their middle points. Place the trammel as shown in the engrav- 
ing, so that the center of the arms shall come directly over the lines. First place the rod along the line A B, 
so that the pencil or point I shall coincide with either A or B. Then place the pin G directly over the inter- 
section of A B and C D. Next place the rod along the line C D, bringing the pencil or point I to either C or 
D, and put the pin H over the intersection of A B and C D. The instrument is then ready for use, and the 
curve is described by the pencil I moved by the hand, but controlled by the pins working in the grooves. 

385. An Improvised Trammel. —It frequently happens that an ellipse is wanted of specified dimensions, 
under conditions which make the use of a trammel desirable. When a trammel is not convenient, a very 
fair substitute is afforded by the use 
of a common steel square and a thin 
strip of wood, like a lath. This 
method of drawing an ellipse is 
also quite useful under ordinary 
circumstances when only a part of 
the figure is required for use, as in 
the shape of the top of a window 
frame to which a cap is to be fitted, 
in which half of the figure would 
be employed, or in the shaping of 
a member of a molding in which a 
quarter, or less than quarter, of the figure would be used. In presenting the rule, we show how to produce the 
complete figure, but the application of it to the other purposes cited is so self-evident that no difficulty can 
arise which would require special explanation. 

386. To Draw an Ellipse of Given Dimensions by Means of a Square and a Strip of Wood. — In Fig. 
228, set off the length of the figure, and at right angles to it, through its middle point, draw a line representing 
the width of the figure. Place a square as shown by A E C, its inner edge corresponding to the lines.- Lay 
the strip of wood as shown by F E, putting a pencil at the point F, corresponding to one end of the figure, and 
a pin at E, corresponding to the inner angle of the square. Then place the stick across the figure, as shown in 
Fig. 229, making the pencil, F, correspond with one side of the figure, and put a pin at G, corresponding with 
the inner angle of the square. In drawing the figure the square must be changed in position for each quarter 
of the curve. As shown in the engravings, it is correct for the quarter of the curve represented by F D. It 
must be changed for each of the other sections, its inner edge being brought against the lines each time, as 
shown. 

3S7. Still another definition of an ellipse is that " it is a figure bounded by a regular curve, which corre- 
sponds to an oblique projection of a circle." 

388. An oblique projection of a circle, perhaps, will be most readily understood if explained by referring 
to a cylinder, as a piece of stove pipe, for example. If the piece of pipe is cut square across and the end 
placed upon a board, and we scribe a line around it, the resulting figure will be a circle. If we now cut the pipe 
obliquely, as, for example, to make a square elbow, or any elbow for that matter — for the angle of the oblique 
cut does not affect the principle at all, it only modifies the proportions of the figure — and we place the end 
thus cut upon a board and scribe around it, as mentioned in the first case, the figure drawn will be an ellipse. 
We have thus, by rough mechanical means, produced what is technically known as an oblique projection of a 
circle, and which by our definition is the figure to which an ellipse corresponds. What we have here done 





Figs. 228 and 229. — To Draw an Ellipse of Given Dimensions by Means of a Square 

and a Strip of Wood. 



58 



Geometrical Problems. 




mechanically may be also accomplished upon the drawing board in a very simple and expeditious manner. The 
demonstration which follows is of especial interest to the pattern cutter, because the principles involved in it 
lie at the root of many practical operations which he is called upon to perform. For example, the shape to cut a 
piece to stop up the end of a pipe or tube which is not cut square across, and the shape to cut the hole in a piece 

which is to fit around a pipe passing through it at other than a right 
angle, like a flange to fit a pipe passing through the slope of a roof and 
other similar requirements of almost daily occurrence, depend entirely 
upon the principles which we shall here explain. "With reference to 
such problems, an ellipse may be defined as an oblique section of a cylin- 
der, the method of drawing the shape of which is given below. 

3S9. To Describe the Form or Shape of an Oblique Section of a 
Cylinder, or to Draw an Ellipse as the Oblique Projection of a Circle. 
— The two propositions which are stated above are virtually one and the 
same so far as concerns the pattern cutter, and they may be made quite 
the same so far as a demonstration is concerned. We confine our expla- 
nation of the engraving to the idea of the cylinder, believing it in that 
shape to be of more practical service to the readers of this book than in 
any other. In Fig. 230, let G E F II represent any cylinder, and A B C D 
the plan of the same. Let I K represent the line of any oblique cut to 
be made in the cylinder. It is required to draw the shape of the pipe as 
it would appear when cut in two by the line I K, and either piece placed 
with the end I K flat upon paper and a line scribed around it. Divide one- 
half of the plan ABC into any convenient number of equal parts, as 
shown by the figures 1, 2, 3, 4, etc. Through these points and at right angles 
to the diameter A C, draw lines as shown, cutting the opposite side of the 
circle. Also continue these lines 
upward until they cut the oblique 
line I K, as shown by l 1 , 2 1 , 3 l , 
etc. In order to avoid confusion 
of lines, draw a duplicate of I K 
to one side, as I' K 1 , making it 
parallel to I K for convenience in 



Fig. 230. — To Describe the Form or Shape 
of an Oblique Section of a Cylinder, or 
to Draw an Ellipse as the Oblique Pro- 
jection of a Circle. 



transferrins 



"With the 



spaces. w mi 

T-square set at right angles to I K, and brought successively against the 
points in it, draw lines through I 1 K', as shown by l 2 , 2 2 , 3 2 , etc. With 
the dividers take the distance across the plan A B C D on each of the 
several lines drawn through it, and set the same distance off on corres- 
ponding lines drawn through I 1 K\ In other words, taking A C as the 
base for measurement in the one case and I 1 K 1 the base of measurement 
in the other, set off on the latter, on each side, the same length as the 
several lines measure on each side of A C. Make 2 : equal to 2, and 3 2 
equal to 3, and so on. Through the points thus obtained, trace a line, as 
shown by I 1 M K? and the opposite side, thus completing the figure. 

390. Another definition of the ellipse is that " it is a figure bounded 
by a regular curve, corresponding to an oblique section of a cone through 
its opposite sides." It is this definition of the ellipse that classes it among 
what are known as conic sections. It is generally a matter of surprise to 
students to find that an oblique section of a cylinder, and an oblique sec- 
tion of a cone through its opposite sides, produce the same figure, but 
such is the case. The method of drawing an ellipse upon this definition 
of it is given in the following demonstration. 




Fig. 231. — To Describe 
Oblique Section of a 
Opposite Sides, or to 



the Shape of an 
Cone through its 
Draw an Ellipse 



as a Section of a Cone. 

^.w. ^ „^ xuiiunuig uomuuuuiauuu. The principles upon which this rule is based, no less than those 
referred to in the last demonstration, are of especial interest to the pattern cutter, becaiise so many of the 
shapes with which he has to deal owe their origin to the cone. 



Geometrical Problems. 



59 



Through the points 
C 




Fig. 232. — To Construct an Ellipse to 
Given Dimensions by Ihe Use of Two 
Circles and Intersecting Lines. 



391. To Describe the Shape of an Oblique Section of a Cone through its Opposite Sides, or to Draw an 
Ellipse as a Section of a Cone. — In Fig. 231, let B A C represent a cone, of which E D G F is the plan at the 
base. Let II I represent any oblique cut through its opposite sides. Then it is required to draw the shape of 
the section represented by II I, which will be an ellipse. In order to avoid confusion of lines, at any conveni- 
ent place outside of the figure draw a duplicate of H I parallel to it, upon which to construct "the figure 
sought, as H' I'. Divide one-half of the plan, as E D CI, into any convenient number of equal parts, as shown 
by 1, 2, 3, 4, etc. From the center of the plan M draw radial lines to these points. From each of the points 
also erect a perpendicular line, which produce until it cuts the base line B C of the cone. From the base line 
of the cone continue each of these lines toward the apex A, cutting the oblique line II I 
thus obtained in H I, and at right angles to the axis A D of the cone, draw 
lines, as shown by l 1 , 2', 3', I", etc., cutting the opposite sides of the cone. 
From the same points in II I drop lines vertically across the plan, as shown 
by l 3 , 2 3 , 3 3 , 4 3 , etc., and also from the same points in II I, at right angles'to 
it, draw lines cutting H 1 I 1 , as shown by T, 2 2 , 3 2 4 2 , etc., thus transferring 
to it the same divisions as have been given to other parts of the figure. 
After having obtained these several sets of lines in different portions of the 
figure, all of which correspond with each other, the first step is to obtain a 
plan view of the oblique cut, for which we proceed as follows : With the 
dividers take the distance from the axial line A D to one side of the cone, 
either A B or A C, on each of the lines l 1 , 2 1 , 3 1 , 4 1 , etc, and set off like 
distance from the center of the plan M on the corresponding radial lines 1, 
2, 3, I, etc. A line traced through the points thus obtained will give a plan 
view of the oblique cut, as shown by the inner line in the plan. Having 
thus obtained the shape of the oblique cut in plan, and having previously 
drawn lines across the plan representing the divisions in H I, the next step is to set off the width of the plan 
at the several points represented by these cross lines upon the lines drawn through II' I'. With E G as a basis 
of measurement, with the dividers take the distance on each of the several cross lines 2 3 , 3 3 , 4 3 , 5 3 , etc., from 
E G to one side of the plan of the oblique cat just described, and set off the same distance on each side of 
H 1 I 1 on the corresponding lines. A line traced through the points thus obtained will be an ellipse. 

392. To Construct an Ellipse to Given Dimensions by the Use of Two Circles and Intersecting Lines. — 
In Fig. 232, let it be required to construct an ellipse, the length of which shall equal A B and the width of 
which shall equal II F. Draw A B and H F at right angles, intersecting at their middle points, K. From K 
as center, and with one-half of the length A B as radius, describe the circle A C B D. From K as center, and 

with one-half of the width H F as radius, describe the circle E F G H. 
Divide the larger circle into any convenient number of equal parts, as 
shown by the small figures 1, 2, 3, 4, etc. Divide the smaller circle into 
the same number of equal and corresponding parts, as also shown by fig- 
ures. By means of the T-sqnare, from the points in the outer circle draw 
vertical lines, and from points in the inner circle draw horizontal lines, as 
shown, producing them until they intersect the lines first drawn. A line 
traced through these points of intersection will be an ellipse. 

393. To Draw an Ellipse within a Given Rectangle by Means of 
Intersecting Lines. — In Fig. 233, let E D B A be any rectangle within 
which it is required to construct an ellipse. Bisect the end A E, obtain- 
ing the point F, from which erect the perpendicular F G,' dividing the rectangle horizontally into two equal por- 
tions. Bisect the side A B, obtaining the point H, and draw the perpendicular H I, dividing the rectangle ver- 
tically into two equal portions. The lines F G and H I are then the axes of the ellipse. F G represents what 
may be familiarly termed the length of the figure, and II I what may be called the breadth of the figure. 
Divide the spaces F E, F A, G D and G B into any convenient number of equal parts, as shown by the figures 
1, 2, 3. From these points in F E and G D draw lines to I, and from the points in F A and G B draw lines to 
the point H. Divide F C and G C also into the same number of equal parts, as shown by the figures, and 
through each of these points draw lines to both I and H, as indicated. A line traced through the several points 
of intersection between the two sets of lines, as shown in the engraving, will be an ellipse. 



Cy^\\i//v> 





Fig. 233. — To Draiv an Ellipse within a 
Given Rectangle by Means of Intersect- 
ing Lines. 



60 



Geometrical Problems. 




Fig. 234. — To Draw an Approximate 
Ellipse in a Given Rectangle by 
Means of Intersecting Lines. 



394. To Draw an Approximate Ellipse in a Given Rectangle oy Means of Intersecting Lines. — In Fig. 
234, let F G H E be any rectangle, within which it is required to draw a figure which shall approximate an 

ellipse in shape, and which shall give the largest surface within the boundary 
of the figure consistent with easy curves. Divide the rectangle into four 
equal portions by the lines A B and C D, as shown. Divide each half of each 
end into any convenient number of equal parts, as shown by the figures. 
Divide each half of each side into the same number of equal parts. Then 
draw the intersecting lines, as shown. Commencing at D, connect with 9, 
1 with S, 2 with 7, 3 with 6, 4 with 5, and so on. A line traced through the 
several points of intersection will be the figure souo-ht. 

395. To Draw an Elliptical Figure with the Compasses, the Length only 
I. inij Given. — In Fig. 235, let A C be any length to which it is desired to 
draw an elliptical figure. Divide A C into 
four equal parts. From 3 as center, and 
with 3 1 as radius, strike the arc BID, and from 1 as center, and with the 
same radius, strike the arc B 3 D, intersecting the arc first struck in the 
points B and D. From B, through the points 1 and 3, draw the lines B E 
and B F indefinitely, and from D, in like manner, draw the lines D G and 
D H. From the point 1 as center, and with 1 A as radius, strike the arc 
E G, and from 3 as center, with the same radius, or, what is equivalent, witb 
3 C as radius, strike the arc H F. From D as center, with radius D G, 
striko the arc G H, and from B as center, with the same radius, or, what is 
equivalent, with B A as radius, strike the arc E F, thus completing the 

figure. 

396. A figure of different propor- 
tions may be drawn in the same gen- 
eral manner as follows : Divide the length A C into four equal parts, as 
indicated in Fig. 236. From 2 as center, and with 2 1 as radius, strike 
the circle 1 E 3 F. Bisect the given length A C by the line B D, as 
shown, cutting the circle in the points E and F. From E, through the 
points 1 and 3, draw the lines E G and E II indefinitely, and from F, 
through the same points, draw similar lines, F I and F K. From 1 as 
center, and with 1 A as radius, strike the arc I A G, and from 3 as cen- 
ter, with equal radius, strike the 





W //' 



Fig. 235. — To Draw an Elliptical Fig- 
ure with the Comp)asscs } the Length 
only being Given. 



Fig. 236. — To Draw an Elliptical Figure 
with the Compasses, the Length only be- 
ing Given. — Another Method. 



arc Iv C H. From E as center, 

and with radius E G, strike the 
arc G D H, and from F as center, with corresponding radius, strike 
the arc I B Iv, thus completing the fignre. 

397. To Draw an Approximate Ellipse with the Compasses to 
Given Dimensions, Using Two Sets of Centers. — First Method. — In 
Fig. 237, let A B represent the length of the required figure and D E 
its width. Draw A B and D E at right angles to each other, and inter- 
secting at their middle points. At the point A erect the perpendicular 
A F, and in length make it equal to C D. Bisect A F, obtaining the 
point N. Draw N D. From F draw a line to E, as shown, cutting 
1ST D in the point G. Bisect the line G D by the line H I, perpendic- 
ular to G D and meeting D E in the point I. In the same manner 
draw lines corresponding to G I, as shown by L I, M O and B O. 
From I and O as centers, and with I G as radius, strike the arcs G D L 
and M E B, and from K and P as centers, with K G as radius, strike the arcs GAM and L B B, thus com- 
pleting the figure. 

398. To Draw an Approximate Ellipse with the Compasses to Given Dimensions, Using two Sets of 
Centers.— Second Method.— In Fig. 238, let C D represent the length of a required ellipse and A B the width. 




Fig. 237. — To Draw an Approximate Ellipse 
with the Compasses to Given Dimensions, 
Using Two Sets of Centers. — First Method. 



Geometrical Prdble 



ins. 



61 





Fig. 23S. — To Draw an Approximate 
Ellipse with the Compasses to Given 
Dimensions, Using Two Sets of Cen- 
ters. — Second Method. 



Lay off these two dimensions at right angles to each other, as shown. On C D lay off a space equal to the 

width of the required figure, as shown by D E. Divide the remainder of D C, or 'the space E C into three 

equal parts, as shown in the cut. With a radius equal to two of these parts, and 

from E as center, strike the circle G S F T. Then with F as center, and F G 

as radius, and with G as center, and G F as radius, strike the arcs, as shown, 

intersecting upon A B prolonged at and P. From O, through the points G 

and F, draw L and O M, and likewise from P, through the same points, 

draw P K and P K From O as center, with A as radius, strike the arc 

L M, and with the same radius, and P as center, strike the arc K K From F 

and G as centers, and with F D and G C as radii, strike the arcs N ]\f and K L 

respectively, thus completing the figure. 

399. To Draw an Approximate Ellipse with the Compasses to Ovoen 
Dimensions, Using Three Sets of Centers — In Fig. 239, let A B represent the 

,: length of the required figure and I> E 

the width. Draw A B and D E at right 

angles to each other, intersecting at their middle points, as shown at 
C. From the point A draw A F, perpendicular to A B, and in length 
equal to D. Join the points F and D, as shown. Divide A F into 
three equal parts, thus obtaining the points Z and I, and draw the lines 
Z D and I D. Divide A C into three equal parts, as shown by Y and 
G, and draw E G and E Y, prolonging them until they intersect with 
Z D and I D respectively, in the points II and J. Bisect J D, and draw 
K L perpendicular to its central point, intersecting D E prolonged in the 
point L. Draw J L and II J. Bisect PI J, and draw M 1ST perpendicu- 
lar to its central point, meeting J L in K Draw N" H, cutting A B in 
in the point 0. L then is the cen- 
ter of the arc J D P, N" is the cen- 
ter of the arc H J, and O is the cen- 
ter of the arc II A B. The points S and U, corresponding to N" and O, 
from which to strike the remainder of the upper part of the figure, may 
be obtained by measurement, as indicated. Having drawn so much of 
the figure as can be struck from these centers, set the dividers to the dis- 
tance L P or L J. By placing one point at E, the remaining center will 
be at the other point of the dividers, in the line E D prolonged, as shown 
byX. 

400. To Find the Centers and True Axes of an Ellipse. — In Fig. 
240, let !M B O B, be any ellipse, of which it is required to find the cen- 
ter and the two axes. Through the ellipse draw any lines, A B and D E, 

parallel to each other. Bisect these two lines and draw F G, prolonging 
it until it meets the sides of the ellipse in the points H and I. Bisect 
the line H I, obtaining the point C. From as center, with any con- 
venient radius, describe the arc K L M, cutting the sides of the ellipse at 
the points K and M. Join K and M by a straight line, as shown. Bisect 
M K by the line 1ST O, perpendicular to it. Through C, which will also 
be found to be the center of N" 0, draw P R, perpendicular to !N" O and 
parallel to K M. Then 1ST and P R are the axes of the ellijise and C 
the point of intersection or center. 

401. In a Given Ellipse, to Find Centers oy which an Approximate 

Figure may he Constructed. — In Fig. 241, let A E B D be any ellipse, in 

which it is required to find centers by which an approximate figure may 

be drawn with the compasses. Draw the axes A B and E D. From the 

A B, and make it equal to C E. Join F E. Divide A F into as many 

In this instance we have determined upo:i 



Fig. 239. — To Draw an Approximate El- 
lipse with the Compasses to Given Dimen- 
sions, Using Three Sets of Centers. 




Fig. 240. — To Find the Centers and True 
Axes of an Ellipse. 




Fig. 241. — In a Given Ellipse, to Find Cen- 
ters by which an Approximate Figure 
may be Constructed. 



point A draw A F, perpendicular to 

equal parts as it is desired to have sets of centers for the figure 



62 



Geometrical Problems. 



four. Therefore, A F is divided into four equal parts, as shown by P O G. Divide A C into the same num- 
ber of equal parts, as shown by K S T. From the points of division in A F draw lines to E. From D draw 

lines passing through the divisions in A C, prolonging them until they inter- 
sect the lines drawn from A F to E, as shown by D U, D Y and D "W". Draw 
the chords U V, V "W and ~W E, and from the center of each erect a perpen- 
dicular, which prolong until they meet other lines, as shown. Thus, commenc- 
ing at the toj>, the perpendicular to W E reaches to the point D ; that to W V 
intersects the line 'W D in the point K, and that to U V meets the line V K 
in the point L. Draw IT L, cutting A C in the point S. Then D is the cen- 
ter of the arc E "W, K is the center of the arc "W V, L is the center of the arc 
V U, and S is the center of the arc TJ N. By these centers it will be seen 
that one-quarter of the figure (A to E) may be struck. By measurement, cor- 
responding points may be located in other portions of the figure. 

402. To Draw an Egg-Shaped or Oval Figure.— In Fig. 242, let A D be 
the required width. Upon A D describe the circle A B D E. From the cen- 
ter of this circle draw C E, at right angles to A D, cutting the circle in the 
point E. Draw D E and A E, and prolong them in the direction of G and F 
respectively. From A as center, and with A D as radius, describe the arc D F. From D as center, and with 
the same radius, describe the arc A G. From E as center, and with E G as radius, complete the figure, as 
shown. 




Fig. 



242. — To Draw an Egg-Shaped 
or Oval Figure. 



63 



THE ART AND SCIENCE OF PATTERN CUTTING. 



403. Before introducing pattern problems, it is appropriate that we should give some attention to the art 
and science of pattern cutting, in order that the reasons for the steps taken in the demonstrations following 
and the directions for the use of tools which are occasionally introduced, may be readily understood. Under- 
lying the entire range of problems peculiar to sheet-metal work, are certain fundamental principles, which 
when thoroughly understood, make plain and simple that which otherwise would appear arbitrary, if not actually 
mysterious. So true is this, that we risk nothing in asserting that any one who thoroughly comprehends all the 
steps in connection with cutting a simple square miter, is able to cut any miter whatsoever. Since almost any 
one can cut a square miter, the question at once arises, in view of this statement, why is it that he cannot cut a 
raking miter, or a pinnacle miter, or any other equally hard form ? The answer is, because he does not under- 
stand how he cuts the square miter. lie may perform the operation just as he has seen some one else do it, or 
as laid down in some book or paper. lie may produce results entirely satisfactory from a mechanical stand- 
point, but after ail is finished he is not intelligent as to what he has done. He does not comprehend the why 
and wherefore of the steps taken. Hence it is, when he undertakes some other miter, that he finds himself 
deficient. Similar statements with reference to j>atterns of shapes derived from cones, and to each and every 
class of problems in sheet-metal pattern cutting, might be made, all teaching the same lesson, and all illustrating 
the importance of a thorough understanding of ground principles. There is a wide difference between the skill 
that produces a pattern by rote — by a mere effort of the memory — and that which reasons out the successive steps. 
One is worth but very little, while the other renders its possessor independent. It is with a desire to put the 
student in possession of this latter kind of skill, to render him intelligent as to every operation to be performed 
that the present chapter is written. 

404. The forms with which the pattern cutter has to deal, for convenience of description, may be divided 
into two general classes. The first of these we will call forms of parallel lines. It embraces moldings, pipes, 
flat surfaces, &c. The second we will call tapering forms. It comprehends all the shapes derived from cones, 
pyramids, &c. We might introduce a third class, embracing forms which in their characteristics belong to both 
of the other two, but since in pattern cutting such forms are treated as belonging to one or the other of the 
classes named, all necessary analysis is obtained by the divisions specified. For example, a vase, the plan of 
which is octagonal, viewed from one standpoint, belongs to the first class, because the lines of molding running 
around it are parallel, while viewed from another standpoint it seems to belong to the second class, because it is 
pyramidal in shape. It rightfully belongs to the first class, because in developing the patterns the form is 
treated as a molding in which octagon miters occur. 

405. The patterns which arise in forms of the first class are, for the most part, what are known as miters, 
and, so far as principles and methods of developing are concerned, are among the simplest and easiest with which 
the pattern cutter has to deal. The methods of measurement, the use of tools, and the general plan of work in 
cutting miter patterns, are not unlike those used in developing shapes derived from cones, &c, although at first 
thought it would seem that they are totally distinct operations. Accordingly, an exemplification of the processes 
of miter cutting, provided we introduce the reason for every step taken, will also cast some light upon the second 
part of our subject. It is possible, moreover, to consider all shapes miters, and to treat everything in the 
6ame general way as moldings. While we shall follow this idea in part, for the sake of better explaining the 



64 



The Art and Science of Pattern Gutting. 




Fig. 243. — Profile 
of a Molding. 

cones. 




various steps taken, we shall take up the second class afterward, and give special explanations of the principles 
upon which its forms depend. 

406. Although the shapes entering into tinware, by daily contact and long association, come to look simple, 
they are in reality the most difficult, in the matter of the development of their surfaces, with which the pattern 

cutter has to deal. On the other hand, moldings, the forms with which cornice makers deal 
almost exclusively, appear to those not conversant with that trade as very difficult indeed. It 
is necessary to divest the reader's mind of these .ideas, in order to prepare him for that form of 
explanation which seems most desirable to introduce in this connection. We shall attempt to 
make clear the science of pattern cutting, first by a familiar talk about moldings, and afterward 
by a similar consideration of cones. The student, therefore, must cease to think that moldings 
are necessarily difficult forms. Although he may not be acquainted with cornice work, he will 
have no difficulty in understanding what we have to say. By familiarizing himself with what 
follows about moldings, he will be the better prepared to understand what we shall say about 
If necessary to his comprehension of moldings and their miters, the experiments herein described should 
be patiently worked out. The encouragement to painstaking effort at this stage is the assurance that a thorough 
understanding of ground principles will make the student 'independent of all examples and 
precedents. It will enable him to formulate his own rules as occasion may require. 

407. Since in sheet-metal work a molding is made by bending the sheet until it fits a given 
stay, a molding may be defined as a succession of parallel forms or bends made to a given stay, 
and, so far as the mechanic is concerned, any continuous form or arrangement of parallel con- 
tinuous forms, made for any purpose whatever, may be considered a molding and treated as 
euch in all the operations of pattern cutting. Keeping in mind, therefore, this fact, that almost 
any shape may be considered a molding so far as the method of obtaining its pattern is con- Fk J- 2 44— a stay. 
cerned, let us examine the nature of moldings and the joints occurring in them, commonly called miters. 

408. A molding may be described as a form or surface generated by a profile passed in a straight or 
curved line from one point to another, this profile being the shape that would be seen 
when looking at the end if the molding were cut off square. Let us consider this definition 
in the light of a familiar illustration. In Fig. 213, let the form shown be the profile of 
some molding. If we cut the shape out of tin plate or sheet iron, as shown in Fig. 
244, it is called a stay. For our purpose, as will appear further on, we require the reverse 
of the stay shown in Fig. 244, or, in other words, the piece cut from the face of the shape 
represented in that figure, which is shown in Fig. 245. 

409. Having provided ourselves with a reverse stay, or " outside stay," as it is sometimes 
called, as shown in Fig. 245, let us take some plastic material— as, for instance, wax or pot- 
ter's clay — and, placing it against a smooth surface, as of a board, move this reverse stay 
along its face until we obtain a continuous form in the clay corresponding to the reverse 

stay, all as illustrated in Fig. 246. By this operation we will have produced a molding in accordance with our 

definition. Our purpose in introducing this illustration is to show more clearly than we are able otherwise the 

principles upon which moldings — 

and, for that matter, all irregular 

surfaces — are measured in the pro- 
cess of pattern cutting; therefore, 

let us carry this same operation a 

step further. 

410. Suppose that the form 

illustrated in Fig. 246 be completed, 

and that both ends of the molding 

be cut off square. It is evident, 

upon inspection, that the length of 

a piece of sheet metal necessary to 

form a covering to this molding- 
will be the length of the molding 

itself, and that the width of the piece will be equal to the distance obtained by measuring around the face of 




Fig. 245. — A Re- 
verse Stay. 




Fig. 246. — Developing a Molding in a Plastic Material, like Clay, by Means of a 

Reverse Stay. 



The Art and Science of Pattern Gutting. 65 

the stay which was used in giving shape to the molding. Keeping this in mind, let us see what we must do in 
order to obtain a covering for it if one end is cut off obliquely. With a thin-bladed knife, or by means of a 
piece of fine wire stretched tight, let us cut off, at any angle, one end of the clay molding which we have con- 
structed. By inspection of the form when thus cut, as clearly shown in the upper part of Fig. 247, it is evident 
that we must have such a shape to the end of the pattern as will make it correspond to the oblique end of the 
molding. 

411. To cut such a pattern as we have just described by a straight line drawn from a point corresponding to 
the end of the longer side of the mold, 



to a point corresponding to the end 
of the shorter side of it, would not be 
right, evidently, because certain parts 
of the covering, "when formed up, fold 
down into the angles of the molding, 
and therefore would require to be 
either longer or shorter, as the case 
might be, than if cut straight, as we 
have stqmosed. It is plain, then, that 
we must devise some plan by which 
measurements can be taken in all these 
angles, and at as many intermediate 
points as may be necessary, in order to 
obtain the right length at all points 
throughout its width. It is easy to 
measure the length of the molding in 
the lines of the several angles, and we 
can also readily obtain measurements 
at as many intermediate points as we 
require, by a simple plan. 

412. Divide the curved parts of 
the stay into any convenient number 
of equal parts, and at each division 
cut a notch, or affix a point to it. He- 
place the stay in the position it occu- 
pied in producing the molding, and 
pass it over the entire length of the 
molding. The points fastened to the 
stay will then leave tracks or lines 




Fig. 247. — The Use of Lines in Laying Off the Pattern of a Covering for a Molding. 



upon the surface of the molding. Now, by measurements upon these lines, the length of the molding at all of 
the several points established in the stay may be obtained. All this is clearly illustrated in Fig. 247. In the 
upper right hand corner of the illustration is shown the stay prepared with points. By moving it as described, 
lines are left upon the face of the molding, as shown to the left. 

413. Now, if we take a sheet of paper, and upon any part of it draw a straight line, as shown by A B in 
Fig. 247, and upon that line set off with the dividers the width of each sjmce or part of the jn-ofile of the stay 
■ — that is, make the space 1 2 in the line A B equal to the space 1 2 in the profile, and 2 3 in the line A B equal 
to 2 3 of the profile, and so continue until all the spaces are transferred — and from the points thus obtained in 
A B draw lines at right angles to it indefinitely, we shall have lines upon the paper corresponding to the lines 
upon the clay molding made by the points fastened to the stay. Next, if we measure the molding upon each 
of the lines drawn upon it, and set off the same length upon the lines drawn upon the paper, we shall obtain 
points through which a line may be traced which will correspond to the oblique end of the molding. There- 
fore we set off, on the line 1 from A B, the length of the molding, measured from its straight end to its oblique 
end, upon the corresponding line upon its face, and upon each of the other lines on the paper the length of the 
molding on the corresponding line on its face. By this means we obtain points, through which, if a line be 
traced, as shown by D, the pattern of the covering will be described. The line A B, laid down by measuring 



66 



The Art and Science of Pattern fritting. 



from the profile, is called the "stretchout line," and the lines drawn through the points in it at right angles to 
it are called "measuring lines." 

414. Now, what we have done in Fig. 247 illustrates what is called " miter cutting." The strict definition of 
the word miter, is the joint between two moldings of like profile at any angle ; but in sheet-metal work it has 
come to mean the shape of the end of a molding or other form required to make it fit against any surface, reg- 
ular or irregular, at any angle. Miter cutting, then, consists of describing in the flat the shape of a gi-ven form 
required to fit against a given surface at a given angle. In this sense almost all patterns are miter patterns. 

415. What we have obtained in Fig. 247, hy means of a clay model — that is, what we have obtained in the 

way of the pattern shown in the 
lower part of the figure, measure- 
ments for which were obtained from 
the lines drawn on the surface of 
the clay model — may be obtained 
just as well hy a drawing. The 
question then is, how can we obtain, 
by lines drawn upon a flat surface, 
the same results as are obtained by 
measurements on lines drawn along 
the surface of a molding? 

416. In moving the profile along 
the clay molding, certain lines were 
made by means of the points af- 
fixed. If the reader will carefully 
examine Fig. 247, he will doubtless 
notice that the lines upon the mold- 
ing made by this means correspond- 
ed in number and position with the 
points in the profile when it is laid 
flat on its side. Hence, if we draw 
the stay or profile, and also repre- 
sent the molding by lines, we are 
able to accomplish the same ends, 
care only being necessary that the 
relative positions of the parts be 
correctly maintained. This is clearly 
illustrated in Fig. 248, which is to 
be compared with Fig. 247. 

417. Let us examine Fig. 248, in 
order to see just what is done to obtain the points of measurement and the dimensions required. First, the 
profile A is drawn in position, as shown. Next, from it a drawing of the required molding is made, as shown 
by FCD6. The rule for drawing the molding and profile may be stated as follows : Place the profile A — 
which, for the sake of comparison, may be a duplicate of the stay used in the preceding illustration, including all 
the intermediate points — in line with the space it is desired the elevation of the molding shall occupy. For 
the lines of the molding, use the T-square in the general position shown by B in the engraving, bringing it 
against the several points in A in order to draw the lines. Draw a line for each of the angles in A, and also 
one corresponding to each of the intermediate points in the stay. Draw the line F G, representing the oblique 
cut, and the line C D, representing the straight end. Then it will be seen that F C D G of Fig. 248, so far as 
lines are concerned, is exactly the same as the molding we made of clay, shown in Fig. 247. The line F G, by 
the definition of a miter, is the " miter line " of this molding. It represents the surface against which the mold- 
ing is supposed to fit. Next lay off a stretchout of the profile A, in the same manner as described in connection 
with Fig. 247, all as shown by II K in Fig. 24S, through the points in which draw measuring lines at right 
angles to it, or, what is the same, parallel to the lines of the moldings. In length make them equal to the length 
of the molding measured upon the coi-responding lines in C D G F. 





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Fig. 248. — Obtaining the Lines of Measurement for the Covering of a Molding by Means 
of a Drawing. Also Illustrating the. Use of the J-Square in Miter Cutting. 



/ 



The Art and Science of Pattern Cutting. 67 

418. Now, if we proceed as suggested in the previous illustration — that is, by using a pair of dividers to 
measure the length of the molding on the several lines, from C D to F G — and if we set off like lengths on 
corresponding lines drawn from the stretchout H K, we will obtain a pattern in all respects correspondin<r to 
the pattern shown in Fig. 247, already referred to. By inspection of the result thus obtained, however, it will 
be seen that the same thing may bo accomplished by using the T-square, as shown by the dotted lines in Fie;. 
248. Therefore proceed as follows : Place the T-square as shown at E, and, bringing it successively against the 
points in F G, established by the lines drawn from the profile A, cut corresponding measuring lines drawn from 
the stretchout II K. Then a line traced through the points of intersection thus obtained, as shown by L M, will 
be the shape of the pattern corresponding to the miter line F G. By this illustration it is evident that the 
T-square may be used with great advantage in transferring measurements under almost all circumstances. 

419. Since we no longer use the dividers to locate the points in the patterns, the position of the stretchout 
line may be taken at will. For convenience, it should be placed as near to the miter line as possible. Hence, 
in practical work, supposing that the molding represented by F C D G is not a very short piece, the stretchout 
line, instead of being opposite the end C D, would be placed somewhere 
near the line of the blade of the T" sc l nare when in the position shown 
by E. We purposely except short pieces of moldings, for the advantage of 
describing the pattern at one operation in such cases sometimes overcomes 
the advantage of placing the stretchout close to the miter line. 

420. By further inspection of Fig. 248, it will be seen that, instead of 
drawing the lines from the points in the profile A the entire length of the 
molding, as there shown, all that is necessary to the operation is short lines 
corresponding to the points of the profile, and extending across the miter 
line F G. The use of these lines, it is evident, is only to locate intersections 
upon the miter line. In other words, all we need is the points in the profile 
A transferred to the miter line F G. The operation of transferring these 
points by short lines, as above described, is termed " dropping the points " 
from the profile to the miter line. 

421. If, instead of the molding terminating against a plane surface, as 
shown by F G in Fig. 248, it be required to develop the pattern to fit 
against an irregular surface, we proceed in exactly the same manner, simply 
substituting for the straight line F G a representation of that surface. From 
this it will be seen that all that is required to develop the pattern of any 
miter, is that a correct representation of the molding be made, showing the 
angle of the miter, and that a profile be so drawn that it shall be in line 
with the elevation of the molding — its face being so placed as to agree with the face of the molding — and that 
points from the subdivisions of the profile be carried parallel to the molding, their intersections with the miter 
line being marked by short lines. 

422. In order to more clearly indicate the point we desire to make by this summary of requirements, let 
us suppose that we have two pieces of molding made of wood, and that we cut the required miter on them by 
means of a saw, and then place them together, as shown in Fig. 249. Now, if we take a piece of sheet iron, 
for example, and slip it into the joint, as shown by A B, and then remove one arm of the miter, we readily see 
that what we have left is exactly what we had in Fig. 24S. In other words, it amounts to a molding fitting 
against a plane, and, hence, the operation of cutting the pattern in such a case as shown in Fig. 249 is identical 
with that described in Figs. 247 and 248. 

423. From all this it is plain to be seen that the central idea in miter cutting is to bring the points from 
the profile against the miter line, no matter what may be its shape or position. Inasmuch as all moldings, if they 
do not member or miter with duplicates of themselves, must either terminate square or against some dissimilar 
profile, it follows that the two illustrations given cover in principle the entire catalogue of miters. 

424. As we remarked at the outset, all patterns may be, in one sense or another, considered miter patterns. 
The principles we have here explained are the fundamental principles in the art of pattern cutting, and their 
application is universal in sheet-metal work. It would be difficult to compile a complete list of miter problems. 
New combinations of shapes and new conditions are continually arising. The best that can be done, therefore, 
in a book of this character, is to present a selection of problems calculated to show the most common applica- 




Fig. 249. — The Cut in each Arm of the 
Molding required to Unite them in the 
Joint A B, is the same as though each 
piece was Calculated to Fit against a 
Plane Surface represented by A B. 



68 



The Art and Science of Pattern Cutting. 



tions of principles which, carefully studied, will so familiarize the student with them that he will have no diffi- 
culty afterward in working out the patterns for whatever shapes may come up in his practice, whether they be 
of those specifically illustrated or not. 

425. From what has preceded we derive the following summary of requirements, together with a general 
rule for cutting all patterns whatsoever: Requirements. — There must be a plan, elevation or other view of the 
shape, in line with its profile, showing the line of the surface against which it miters. 

426. Hide. — 1. Place a stretchout of the profile on a line at right angles to the direction of the molding or 
other shape, as shown by the plan, elevation or other view, and draw measuring lines parallel to the molding. 
2. Drop points from the profile to the miter line or line of joint, carrying them in the direction of the molding 
or other surface. 3. Drop the points thus obtained from the miter or joint-line on to the measuring lines of the 
stretchout, at right angles to the direction of the molding or surface. 

427. The student who gives careful attention to these rules will at once remark that the operation of cut- 
ting a common square miter — that is, a square mitei between the moldings running across two adjacent sides of 

a building, for example — does not employ a miter line, and therefore 
appears to be an exception. Yet we have remarked (Section 403) 
that a thorough understanding of how a square miter is cut comprehends 
within itself the entire science of pattern cutting. It is because a square 
return miter — for such is the distinctive name applied to the kind of square 
miter in question — is in one sense an exception to the general rule, that it 
is so valuable for the purposes of illustration. A miter of this kind 
admits of an abbreviated method. The short rule for cutting it is usually 
the first thing a pattern cutter learns, and the operation is very generally 
explained to him without any reason being given for the several steps 
taken. In many cases it would bother him to cut the pattern by any 
other than the short method, even after he has obtained considerable pro- 
ficiency in his art. Hence it is that, to all who have any previous knowl- 
edge of pattern cutting, the rules above set forth seem inadequate, or, to 
put it otherwise, a formula to which there are exceptions. 

428. To clear up these doubts in the mind of the student, we will 
first introduce an illustration of the short method of cutting a square 
return miter, and afterward we will show the long method, or the plan 
which is in strict accordance with the rule above given, combined with 
the short method, thus showing the relationship and correspondence 
between the two. 

429. Fig. 250 shows the nsual method of developing a square return 
miter, being that in which no plan line is employed. The profile A B is 
divided into any convenient number of spaces, as indicated by the small 

figures in the engraving. The stretchout E F is laid off at right angles to the lines of the moldings, and, through 
the points in it, measuring lines are drawn parallel to the lines of moldings. From the points established in 
the profile, lines are dropped cutting corresponding measuring fines. Then the pattern is obtained by tracing a 
line through these points of intersection. 

430. In this operation it will be noticed that we have fully complied with the stipulations of the first rule 
given. "We have placed the stretchout at right angles to the lines of the molding, and have drawn measuring 
lines parallel to those lines, but when it comes to the second and third parts of the rule it would seem that we 
have clone something else than anticipated therein. We have, apparently, employed no joint line or plan, but 
have dropped points directly from the profile on to the measuring lines. 

431. Let us now examine Fig. 252, which in its upper part contains the short rule just described, and which 
by G F, shows the use of the plan line of the joint or miter line. The pattern, as developed by the long method, 
is shown on the lower portion of the cut to the right. Eeferring to Section 425, it will be seen that we have 
complied with the requirements therein recited. We have a plan of the shape (F G) in line with the profile 
A B. By spacing the profile in the usual manner, and drawing lines from the points in it toward the miter 
line, we have the lines of the molding in plan, at right angles to which, by the first part of the rule in Section 
426, the stretchout is to be placed. Therefore, we lay off C D at right angles to H F, and draw measuring 




Fig. 250. — The Usual Plan of Cutting a 
Square Miter in which no Joint Line is 
used. 





The Art and Science of Pattern Cutting. 69 

lines perpendicular to it, or, what is the same, parallel to the lines of the molding in the plan, as stipulated in 
the rule. We have already dropped points from the profile on to the miter line, as recited in the second part of 
the rule. So there remains only the third part to be complied 
with. Placing the blade of the T-square at right angles to the 
lines of the molding in the plan, and bringing it successively 
against the several points in F G, we cut corresponding meas- 
uring lines drawn through the stretchout. Then a line" traced 
through these points of intersection will be the pattern sought. 

432. Laying off a stretchout below the profile and at right 
angles to it, as shown by C D, through the points in which 
measuring lines are drawn, and tracing a line through the 
points of intersection between corresponding measuring lines 
and lines dropped from the profile, also produces the pattern, as 
shown by C E. This last operation is the short method, or 
the same as shown in Fig. 250. By comparison it will be 
seen that the two patterns C E and C E 1 are identical. 

433. Since the miter line F G bisects the right angle 

II F Iv, the two arms of the 
miter must be identical. 
Hence, all the operations in 
connection with the patterns 
may be performed on one 
side of the line. By com- 
parison it will be seen that 
the relationshirj between C E 
and the miter line, and C E" 
and the line, are the same. 
Dropping points from a pro- 
file against a line inclined 45 degrees, as F G, and thence on to a stretchout, 
gives the same result as dropping them on to the stretchout in the first place. 
Hence it is that the portion of the operation shown in the lower part of the 
engraving may be dispensed with. 

434. A very common mistake made by beginners in attempting to apply 
the general rule for cutting miters given in Section 426, is that of getting the 
miter line in a wrong position with reference to the profile. For example, 
instead of drawing a complete plan, as shown by L H F K M in Fig. 252, by 
which the miter line is located to a certainty, and in connection with which it 

is a simple matter to correctly place 
the profile, it is very customary to 
attempt the operation by drawing the 
miter line only, placing it either above, 
below or to one side of the profile. The 
mistake is made by having the line to 
the side of the profile when it should 
be either above or below it, and vice 
versa. Fig. 251 illustrates a case in 
point. The engraving was made from 
the drawing of a person who attempted 
to cut a square return miter by the 
rule, using a miter line. By placing 
profile, as shown in Fig. 252, a square face miter — for example, 



Fig. 251. — A Square Face or Panel Miter. This Cut 
Illustrates a Mistake often made by Students in 
Attempting to Employ a Miter Line in Gutting Square 
Return Miters. 



6 7 8 9 10 11 it 



Fig. 252. — A Comparison of the Short or Usual Method of Cutting a Square Miter 
with the Long Method, or that which takes all the Steps laid down in the Rule. 



the line E F to the side instead of below the 

such as would be used in the molding running around a panel or a picture frame 

what was desired. 



-was produced in place of 



70 



The Art and Science of Pattern Cutting. 



435. No better rule for avoiding errors of this kind can be given than the exercise of the greatest thought- 
f nlness and care. It is better to draw a complete plan, as shown in Fig. 252, thus demonstrating to a cer- 
tainty the correct relationship of the parts, than to save a little labor and run the risk of error. So far as it is 
possible to formulate a rule for such operations, it may be presented thus: Place the profile, with reference to 
the plan or elevation, so that lines drawn from the points in it will correctly represent the molding in plan or 
elevation, as the case may be. Thus, in Fig. 252, the lines dropped from the profile to the miter line, and thence 
carried to the right, represent the members of the molding as they would appear if we were above it and looked 
down upon and through it. The relative position of the parts is evidently correct for the end in view. Apply- 
ing the same test to Fig. 251, it will be observed that the lines drawn in the plan, or elevation, whichever it 
may be considered, are correct for a square panel miter, but are incorrect for the plan of a square return miter, 
which it was the design of the draftsman to produce when he made the drawing. 

436. Always bear in mind that miter cutting, and for that matter all pattern cutting whatsoever, is simply 
a system of measurements upon surfaces. Of necessity, the surfaces are represented by diagrams in the flat. 
Two or more views are required to obtain the same dimensions from a drawing of an object as would be got 

in one operation from the object 
itself. The two or more views are 
to be so arranged that different por- 
tions are presented at the same time 
in proper combination. "We have 
already seen (Section 417) how, by 
means of a profile and a drawing 
properly placed, the same results 
were accomplished as were obtained 
by measurements upon the molding 
itself. Keep such comparisons in 
mind, and think out what is wanted 
to be done before the drawing is 
commenced. 

437. In further elucidation of 
the principles of miter cutting, and 
as illustrating the directions just 
given, we show in Fig. 253 some 
of the patterns in a very common 
form of window cap. The miters 
illustrated are of the kind called 
" face miters," the one represented by the line C D being a square miter, while that at E F is at some other 
than a right angle. The profile A B is spaced in the usual manner, and lines from the points are carried through 
the various parts composing the cap, parallel to the lines of molding. The stretchout G H is laid off at right 
angles to the lines of that portion of which the pattern is required. The measuring lines being drawn in 
accordance with directions already given, the T-square is placed with the blade at right angles to the lines of the 
molding, and being brought successively against the points in the two rniter lines, the measuring lines of cor- 
responding number are cut. Then lines traced through these points of intersection complete the pattern. 
Had it been desired to obtain the pattern for that portion shown by A C D B in the elevation, a stretchout line 
would have been drawn at right angles to it. The square return miter would be dropj>ed from the profile, 
while the opposite end of the piece would be obtained by dropping points from the miter line D C. 

438. Having now, as we think, made clear the principles of pattern cutting, at least so far as they can be 
illustrated by simple miters, we desire to return again to the rule laid down in Section 426, in order to present 
another conception of a square return miter, which will show that the short method we have taken so much 
pains to explain by comparing it with the long rule, is not so much an exception to the general rule as would at 
first be supposed. Referring to Fig. 250 for illustration, we will apply the rule to the operations there shown. 
In the first place, we place a stretchout, E F, at right angles to the direction of the molding, as shown by 
the elevation, for ACDB represents the molding in elevation. We next draw measuring lines parallel to 
the molding. Thus E G and the lines below it are parallel to A C and B J). The second part of the rule 




PROFILE 



Fig. 253.- 



ELEVATION 



3 i 5 6 7 8 9 10 11 12 



-Patterns in a Common Form of Window Cap, Introduced in Further 
Elucidation of Principles. 



The Art and Science of Pattern Cutting. 



71 



says : " Drop points from the profile to the miter line or line of joint, carrying them in the direction of the 
molding." A B is evidently the profile, from which points are to be dropped on to the miter line, or line of 
joint. A moment's investigation will show that A B is also the miter line in this case. What we really want 
to do is to cut the pattern, to such a shape that, when it is formed up, one end will be straight and the other end 
present the profile shown by A B. This view of the case makes A B the miter line. In connection with our 
description of Figs. 247 and 248, we remarked that if the shape required to be given to the end of the molding 
were other than that represented by a straight line in the elevation (F G, Fig. 218), the operation would still be 
the same, the only change to be made being the substitution of a curved or mixed line in place of the straight 
line. Now, A B of Fig. 250 may be considered a mixed line, substituted for the straight line F G in Fi°\ 248. 



mmmrn^m 





Fig. 254. — The Patterns of an Octagonal Vase Developed by the "[-Square, Introduced to Illustrate the Use 

of that Instrument. 

Therefore, in spacing the profile we also dropped points upon the miter line. Our compliance with the third 
part of the rule is evident without special explanation. By investigations and comparisons of this kind it 
becomes evident that there is a unity of principle underlying all the operations in pattern cutting. If the stu- 
dent is able to grasp and master this central idea, his success as a pattern cutter is assured. 

439. In order to make the use of the T-square for transferring distances and dropping points better 
understood, we present a diagram of the patterns of an octagonal vase, in the development of which this 
instrument plays an important part. Beferring to Fig. 254, it will be seen that the profile C E is drawn directly 
over the plan, and that points from the profile are dropped across so much of the plan as it is necessary to use in 
developing the pattern for one section. For this purpose the T-square is employed in the position shown in the 
engraving. Thus the miter lines G M and G 1ST represent the boundaries of one of the sections in the plan. 
Points from the profile are dropped so as to cut these two lines. It is not necessary to continue them entirely 



72 



The Art arid Science of Pattern Cutting. 




across the plan. Simply crossing the rniter lines with short fine marks answers every requirement. At right 
angles to the side of the vase, as shown in the plan, the stretchout S T is laid off, using the T-square as shown 

by the dotted lines in the engraving. Through the stretch- 
out measuring lines are drawn in the usual manner. By 
bringing the T-square against the several points in the miter 
lines G M and G N, and thus cutting measuring lines of 
corresponding numbers, points of intersection are obtained, 
through which, if lines be traced, the form of the pattern, 
as shown by TJ V X W, will be obtained. 

•±•±0. Before taking up the siibjcct of tapering surfaces, 
we will introduce Fig. 255, which shows patterns of vases, 
the plans of which are various regular polygons, all devel- 
oped from the same profile. This diagram serves to illus- 
trate several points. It shows the relationship of profile 
and plan ; the use of miter lines in the plan, and the appli- 
cation of one general rule to what are ordinarily considered 
separate and distinct problems. It further shows, in part, 
the reason for the assertion made at the commencement of 
this chapter, that proper knowledge of a square miter is 
adequate for cutting any miter. Detailed description of 
the steps shown is not necessary, because they are the same 
as described in connection with the last figure. Problems 
illustrating the same miters are also to be found in their 
proper places in another portion of the book. 

441. A term of somewhat frequent occurrence in geo- 
metrical works is " a solid of revolution," the meaning of 
which, as defined by "Webster, is as follows : " A solid gen- 
erated by the motion of a surface about a line as its center 
or axis." Defined in more familiar terms, it may be de- 
scribed as a solid whose outline corresponds to the form 
described by the rotation of a plane of some defined shape 
around one of its sides. A right cone (see Section 98) is 
one of the most common examples of a solid of revolution. 
Thus, if a right-angled triangle, C E D, Fig. 256, be revolved 
about its altitude, C E, as an axis, the form described by its 
hypothenuse will be a cone. A cylinder, Fig. 257, is another 
example in point. If a rectangle, as shown by C D F E, be 
revolved about one of its sides, C E, as an axis, the form 
generated will be a right cylinder. 

442. Our purpose in introducing this term and these 
illustrations in this connection, is to make clear by contrast 
what cannot be so well shown by other means. We have 
already explained that in sheet-metal pattern cutting all 
objects are treated as solids — that the shell, with which we 
really deal, is considered the envelope stripped from a solid. 
Keeping this in mind, and examining the nature of cones 
and cylinders in the light of the definition above given, the 
reasons for some of the steps taken in developing patterns 
for them at once become apparent. Perhaps, however, we 
can show this better by describing one or two experiments 



Fig. 255. 



-Cutting The Patterns for a 
any Number of Pieces. 



Vase or Urn in 



which may be made with cones, cylinders, etc., just as was shown in connection with moldings where we 
employed the clay form. 

443. Keeping in mind how solids of revolution are generated, let us investigate their properties by some 



The Art and Science of Pattern Cutting. 



73 





Fig. 256.-4 Eight Cone, Generated by 
the Revolution of the Right-. Angled Tri- 
angle C E D about its Alois, C E. 



Fig. 257. — A Right Cylinder, Generated 
by the Revolution of the Rectangle 
C D F E about C E, one of its Sides. 



experiments in the revolution of solids. Let us suppose that we have a cone, a cylinder, a cube, a prism and a 

pyramid, all with their surfaces blackened in such a way as to make an impression or print when they are 

revolved or rolled over a sheet of paper. Commencing with the cone, as 

shown by A B in Fig. 258, we will mark some point in its base by which 

to note how far it has revolved, and will turn it so as to make one com- 
plete revolution. The resulting 
figure, which of course corresponds 
to its surface, is as shown by 
C A' D. The point A, having no 
diameter, remains in one spot dur- 
ing the operation, but the base 
travels the distance shown by the 
curved line C D, or, in other words, 
a distance equal to the circumfer- 
ence of the base. From this sim- 
ple experiment it is easy to see 

what is to be done to work out by lines the pattern of a right cone. 
In the first place, one end of the pattern will be a point like A 1 ; the 
other end, evidently, will be a curve, all points in which are equally dis- 
tant from A 1 , and the length of which is equal to the circumference of 

the base. Therefore we set the dividers to a radius equal to the bight of the cone, and from any point, as A 1 , 

describe an arc, making its length equal to the circumference of the base. 

4-14. Since the distance from the apex of a right cone to its base is the same at all points, the system of 

measurements which we described in connection with the clay molding 

would not seem to apply in problems relating to cones. This, however, 

is not the case. The peculiarity mentioned is incidental to one form 

of the cone alone, and makes abbreviated methods possible with it. We 

could obtain the same results with a right cone by measuring on the sur- 
face instead of revolving it. For 
example, we might have drawn a 
straight line from the apex to the 
base of the cone, and then, meas- 
uring one inch along the base 
in the direction of its circumfer- 
ence, drawn another line to the 
apex, and so continue around it. 
For the pattern shape in this case 
we would have drawn a line equal 
in length to the first line above 
described, and then, measuring from 
one end of it, w T e would have laid 
off an inch, space, drawing another 
line to the end, representing in the 
pattern the apex of the cone, and 

so on. Or, to describe the operation in another way, for the pattern we 
would have proceeded to construct, side by side, a number of triangles 
corresponding to the triangles drawn on the face of the cone. By this 
plan the similarity between the measurements necessary to the develop- 
ment of a cone pattern and those employed in miter patterns is at once 

perceived. The right cone, like the square return miter, admits of a short method, but patterns of other forms 

of cones require measurements somewhat after the plan above outlined, though in many cases much more com- 
plicated. To these we shall give attention further on. 

M5. Fig. 259 represents a similar experiment performed with a scalene cone. In this case the revolution 





Fig. 258. — The Revolution of a Right Cone 
by which the Shape of its Envelope is 
Described. 



Fig. 259. — A Scalene Cone, Revolved in 
such a Way as to Show the Shape of 
its Envelope. 



74 



The Art and Science of Pattern Cutting. 



of the cone is made to begin with tlie shortest point, so that its longest length falls in the middle of the pattern 
surface. By comparing this shape with that last described, it is evident that some such system of measurement 

as alluded to above will be necessary to determine the shape represented by D E F. 

"We shall not stop here to describe how measurements are applied in this case, 

because it will be necessary to take up the same 

subject further on. 

446. In Fig. 260 we show a similar experiment 

with a six-sided pyramid. Here it will be seen that 

the pattern is a succession of triangles, each of 

which is equal to one of the faces of the pyramid. 

The manner of developing a pattern of this kind 

is almost self-evident. By describing an arc of a 

circle from the center A, with a radius equal to the 

length of one of the faces, and then stepping off in 

this arc spaces equal to the width of the faces 

measured at the base, the shape indicated by A' I C 

will be obtained. Supposing that this pyramid 

were cut on the line x y, as shown in Fig. 260, the 

revolution of the solid would give the shape indi- 
cated in Fig. 261. The pattern would be obtained 

by lines and measurements in the same general 

manner. Having established that part of the pat- 
tern corresponding to the base, as described in con- 
nection with the previous illustration, and as indi- 
cated by I, H, G, F and C, and drawn lines to the 

center, it is a simple matter to measure up each 

of the several angle lines a distance equal to the 

hight of the corresponding angle in the solid itself, 

thus determining the broken line which in Fig. 261 represents the top of the pat' 





Fig. 260. — A Hexagonal Pyr- 
amid, Revolved in such a 
manner as to Describe the 



Fig. 261 .—Hexagonal Pyramid, with 
so much of its Apex Removed as 
is indicated by x y in Fig. 260, Re- 
volved so as to show the Shape of 
its Covering. 





Shape of its Covering. 

tern. Several examples illustrating this principle will be found among the pattern problems. 

447. Leavino- the cone for a moment, let us revolve a cylinder in the same general manner as we have been 

describing. The shape produced 
is shown in Fig. 262 by D E G F. 
It is evident that the length D E 
must be equal to the length A B 
of the cylinder, and that the width 
E G must be equal to the circum- 
ference of the cylinder. Suppose 
one end of the cylinder to be cut 
off obliquely, as shown in Fig. 263, 
and that the solid is then revolved 
in the same manner. Here one end 
of the pattern shape is irregular, as 
shown by E F G, and becomes 
what we have already described as 
a miter pattern. Beferring again 
to Fig. 262, if an opening be cut 
in the cylinder, as indicated by 
of the elevation, and it be revolved, 
a form similar to the shape indi- 
cated by C 1 in the pattern will be 
produced. Without describing in 



Fig. 252. — A Cylinder Revolved, shotving 
Vie Shape and Extent of its Covering. 
Also showing the Shape in the Pattern 
of an Opening made in its Side. 




1 

Fig. 263. — A Cylinder with One End Cut 
off Obliquely and Revolved so as to show 
the Shape of its Envelope. 



detail all the features of these experiments, it is evident, we think, that the system of measurement on the sur- 



The Art and Science of Pattern Gutting. 



75 



/ 




w 
B 



Fig. 264. — The Covering of a Triangular Prism, obtained 
by Revolving it as Before Described. 



face of solids, by which the shape of various parts is determined, is the same in all cases, and in principle is 
identical with that described in connection with the clay molding and miter patterns at the commencement of 
this chapter. 

448. Fig. 264 shows the covering of a triangular prism, obtained in the same general maimer as we have 
been describing, but which, it must be evident to the reader, 

can be just as well obtained by lines and measurements. 
Fig. 265 shows the same thing applied to a cube. If the 
student will keep in mind these experiments, and when 
puzzled over difficult problems will picture in his mind the 
form that would be produced by the revolution of the solids 
with which he has to deal, he will find it of great help to 
him in determining the best method of obtaining the lines 
and measurements required. 

449. We have remarked that the most difficult prob- 
lems in pattern cutting relate to conical shapes. Besides 
the right cone, some of the properties of which we have 
just illustrated, there are conical forms whose bases are 
elliptical instead of circular. With such figures the steps 
necessary to develop the shape of the covering are, of course, 
very much more complicated than those employed for the 
simple form we have named. Although it is possible, in 
some instances at least, to revolve the solids we have 
just referred to, the shapes thus produced are so irregular 
in outline as to show at once that quite different means from 
any so far described are necessary to obtain the requisite lines and measurements for developing the pattern. In 
the preceding chapter we referred to some of the properties of the ellipse, showing how it may be pro- 
duced by string and pencil (Section 377), also how approximate figures may be drawn by the compasses from 
two or more sets of centers (Sections 395 to 399), and how an oblique section of a cone through its opposite 
sides and an oblique section through a cylinder both produce this figure. (Section 390.) An infinite variety of 

ellipses is possible, the range being from a close resem- 
blance to a circle on one extreme, to a figure suggesting a 
straight line on the other, and the solids which may be 
erected on the ellipse for a base vary quite as much. 
Since an oblique section through a solid, the base of 
which is a circle, as, for example, a cylinder or a right 
cone, gives an ellipse, it follows that oblique sections 
through solids whose bases are elliptical may produce 
circles. Careful attention as to the nature of the forms 
with which he has to deal, is always required upon the 
part of the pattern cutter. Sometimes it is necessary 
for him to resolve a form into its simple component parts 
before it is jwssible to develop the patterns at all. It is 
only by thoroughly understanding the nature and proper- 
ties of the ellipse and of elliptical solids, so that they are 
recognized in whatever form encountered, that he is ena- 
bled to develop the intricate shapes peculiar to work of 
this kind. 

450. We described in Section 443 a method by which 
the envelope of a right cone may be drawn, deriving the rule from the experiment that had just been made, of 
revolving the solid in a way to show the form of its covering by an impression or print. The pattern of a frus- 
tum of a cone is much more frequently required in sheet-metal work than that of the complete cone. The 
method of proceeding in such cases is very similar to that employed with the complete figure. It is simply 
necessary to restore that portion of the cone cut away, as shown in Fig. 266, and employ two radii of different, 




H 



M 



K 



P R 

Fig. 265. — The Covering of a Cube, developed by Revolv- 
ing it so that its Several Sides come in Contact with the 
Paper. 



76 



The Art and Science of Pattern Cutting. 



length. 



If a pin be fastened at the apex C of a right cone, Fig. 267, and a piece of thread he tied thereto, 
carrying points B and A, corresponding in position to the upper and lower faces of the frustum, and the thread 
being drawn straight be passed around the cone, the points will follow the line of the faces of the frustum 
throughout its course. If we then take the thread and pin from the cone, and fastening the pin as a center 

upon a sheet of paper, as shown in Fig. 268, carry it around the pin, keep- 
ing it stretched all the time, the track of the points fastened to the thread 
will describe the shape of the envelope of the frustum. Ey omitting the line 
produced by the upper of the two points, the envelope of the complete cone will 
be described. In both cases the lengths of the arcs described by the thread and 
the points attached to it are to be governed by the circumference of the base of 
the cone, as we have already described. Our ob- 
ject in introducing this experiment is to show a 
method applicable in common to right and elliptical 
cones, by which the correspondence between these 
two shapes may be the more readily discerned. 

451. Since all the points in the boundary line 
of an elliptical figure are not equally distant from 
one common central point, it follows that the dis- 

Fig. 266. — Frustum of a Right x ,,, . 

Cone. The Dotted Lines show tance from the apex of an elliptical cone to the 

the Cone Restored for the points in its base is a constantly varying one. 

purpose of Pattern Cutting. Therefore quite different means are necessary in 

developing the shape of the envelope of an elliptical cone from those employed 
with the right cone. "VVe will first describe the method of performing this ope- 
ration with lines, and will afterward refer to the parts into which certain ellipti- 
cal cones may be resolved by analysis of their shape, showing in that connection 
the pin and thread method applied to their development, by which comparison 
can be made with what we have already presented. 

452. In Fig. 269 we show the frustum of an elliptical cone, the envelope of 

which is required. The ele- 






( 

Fig. 267. — A Right Cone, loith 
Pin fastened at the Apex, to 
■which is attached a Thread, 
with Points corresponding to 
the Upper and Lower Faces of 
the Frustum. 



vations and plan of the same 
figure are shown in Fig. 270, 
with the necessary lines for developing the pattern. By 
inspection of the two elevations shown in the latter figure, 
it will be seen that the hight of both is the same, speaking 
now more particularly of the full cone. Since the base 
width of the side is greater 
than the base width of the 
end, it follows that the an- 
gle of inclination or slant 
of the side is greater than 



that of the end. Or, by 

inspecting the plan, it will 

be seen that the distance 

B A, representing the flare 
at the ends, is greater than C D, which shows the flare at the sides. By reason 
of this irregularity the outline of the pattern will be a broken line instead of 
a regular curve, and is to be obtained by constructing a number of triangles. ~We can make this plainer, per- 
haps, by referring to the definition of a right cone, which, as already given, is "a solid generated by the revolu- 
tion of a right-angled triangle around its altitude as an axis." Now, an elliptical cone, it is evident, cannot be 
generated by the revolution of a right-angled triangle around an axis, because the points in its base constantly 
vary in their distance from the center, and yet it is evident that a right angle may be constructed whose altitude 
shall be equal to the axis of the elliptical cone, and whose base shall be equal to half of the length of the base, 
and also that a similar triangle may be drawn, the altitude of which shall likewise be equal to the hight 



Fig. 268. — Describing the Pattern of the Envelope of a 
Frustum of a Right Cone by means of a Pin and 
Thread. 




Fig. 269. —The Frustum of an Ellip- 
tical Cone, the Envelope of which 
is to be Described. 



The Art and Science of Pattern Cutting. ffl 

of the axis of the cone, and the base of which shall be equal to half of the width of the base of the cone. 
Such a right-angled triangle as we have been describing is indicated in Fig. 270 by KEF, in which K E 
the hight, is the axis of the cone and E F is one-half of the length of the base, equal to X A of the plan. A 
similar triangle would be composed of K E for alti- 
tude and X D of the plan for base. Herein is a 
suggestion of the means which may be employed 
for describing the envelope of this shape. This 
solid cannot be generated by the revolution of a single 
triangle, but we can construct a number of triangles, 
having varying bases but one common altitude, by 
measurements on which all necessary dimensions 
may be obtained. In Fig. 270, divide one-quarter 
of the plan, as P D, into any convenient number 
of equal spaces, as shown by the figures 1, 2, 3, 4, 
etc. From the points thus established draw lines to 
the center of the plan X, as shown. At any con- 
venient place draw a straight line, M X, as shown 
in Fig. 271, which in length make equal to the hight 
of the cone. At right angles to the base of this 
line lay off X 1, in length equal to X 1 of the plan, 
Fig. 270, and in like manner set off distances equal 
to the length of the several lines drawn from X to 
the boundary of the figure in the plan, and from 
these points draw lines to the point M. By this 
operation we have in one diagram a set of triangles 
corresponding to the lines drawn in the plan. The 




Fig. 270.- 



-Side Elevation, End Elevation and Plan of the Frustum 
shown in the Preceding Figure. 

next step is to apply the dimensions we have now obtained to the development of the pattern. By insj^ection 
it is evident that the points in the boundary of the pattern corresponding to the points 1, 2, 3, 4, etc., in the 
plan, will be the same distance from one common center as these points in base or plan are from the apex of the 
cone. The distance of these points from the apex of the cone is indicated by the hypothenuses of the triangles 
constructed in Fig. 271 . Therefore, taking M as a center, we set the compasses to the several lengths M D, 

M 7, M G, etc., as radii, and describe arcs as shown to the left. At any 
convenient point in the arc corresponding to D, as D 1 , we draw a line to 
M, which will represent one side of the pattern. From D" we then lay 
off the stretchout of the base, as shown by the divisions 1, 2, 3, 4, etc., 
in the plan, taking the distance in the dividers and stepping from one 
arc to another, as shown in Fig. 271. A similar set of arcs is to be 
drawn from the intersections of the line representing the top of the frus- 
tum with the hypothenuses of the triangles. Then lines drawn from the 
points established on the lower set of arcs, will intersect the last arcs 
drawn at points representing the upper line of the pattern. 

453. The method just explained for obtaining the pattern of the 
envelope of an elliptical cone applies to what we may term perfect ellip- 
tical cones only. By a perfect elliptical cone we mean a solid whose base 
is an ellipse and whose apex is a point. Such a solid as we have shown 
in Fig. 270, gives frustums of which the flare at sides and ends is une- 
qual. Patterns of this kind, however, are less frequently required in 
practice than those in which the flare is alike throughout. If the flare of ends and sides is made the same, the 
resulting solid, if we attempt to complete it, will not be a perfect elliptical cone, but rather an irregular form, 
which it will be found, upon careful inspection, can be resolved into several simple parts. In explanation of this 
form we will first describe the usual rule for developing patterns of regular flaring ware, and will afterward 
undertake to explain the reasons for the several steps taken. 

454. In Fig. 272 is shown the usual method employed for describing the patterns of regular flaring ware. 




,*».s.' D x 

Fig. 271. — Diagram of Triangles constructed 
from Measurements upon Elevations and 
Plan in Fig. 270, showing how they are 
Spread in Describing the Pattern. 



78 



The Art and Science of Pattern Cutting. 



KLNM represents the frustum of which the envelope is required. A B C D is the plan of the same on the 
base line, while the inner curve represents the plan of the upper surface of the frustum. The ellipses repre- 
senting the plan of the article, from the requirements of succeeding operations, are struck from centers, or, if 

true elliptical curves are employed, they are to be resolved into arcs of 
circles by the method explained in Section 401. In this case, in order to 
simplify the explanation, we have employed a plan described from two 
sets of centers. Having determined these centers, we draw the lines 
C X, D E, etc. The next operation is to construct the diagram shown in 
the upper part of the figure, which determines the radii to be employed 
in developing the pattern. Lay off O P equal to D E of the plan, the latter 
being the radius of the arc E C W. Upon O erect the perpendicular O J. 
continuing the same in the direction of J indefinitely. Make O S equal 
to the straight bight of the frustum, and draw S U parallel to O P, mak- 
ing it equal in length to D H of the plan, or the radius of the arc H G Y. 
Now, if we draw a straight line through the points P and TJ thus estab- 
lished, and continue the same upward until it meets the perpendicular O J 
in the point J, we shall have a triangle which, if revolved upon its side 
J O as an axis, will generate so much of the conical shape of which the 
frustum in question is a part as corresponds to the ares in the plan struck 
from D as center. Next, if we locate the point P in the base of the 
diagram by making R equal to D F of the plan, and upon P ei - ect the 
perpendicular P Z, producing the same until it meets the line P J in the 
point Z, we shall have in Z P P a triangle, which, if revolved upon 
Z P as an axis, would generate so much of the shape of which the 
frustum K L 5 I is a part as in plan is struck from F as center. 
The succeeding steps are self-evident, and we shall describe them without 
a diagram, because it will be neces- 
sary to show the same thing in its 
proper place among the pattern prob- 
lems. From any convenient point as 
center, with J IT and J P as radii, 
strike parallel curves, in length equal 
toECW and II G V of the plan 
Fig. 21 2.-The Usual Method of Develop- respectively. This will represent so 

ing the Patterns of Regular Flaring much of the envelope as belongs to 

Shapes. the cone which we said would be gen- 

erated by the revolution of the large triangle J O P. From the terminal 
points in these curves draw lines to the center from which they were struck, 
and then with radii Z TJ and Z P, from a center in one of the lines just 
drawn, continue the curves, making the arcs in this case equal in length to 
X A E and the corresponding inner line- of the plan struck from F as cen- 
ter. In this operation we have described the envelope of that part of the 
frustum which, as explained above, is a part of the cone that would be gen- 
erated by the small triangle revolving about its altitude as an axis. These 
steps will give one-half of the pattern, and a repetition of them will pro- 
duce the other half. 

455. "We have indicated, by the method of explanation above employed, 
that the solid of which the frustum K L N M of Fig. 272 is a part, is com- 
posed of portions of right cones of different diameters and altitudes joined together. The reasons of the several 
steps taken will be better understood if we show just what such a solid looks like, and how it is resolved into 
its component elements in the process of pattern cutting. We remarked that it was necessary, on account of 
subsequent operations, to employ a plan struck from centers, and then, having determined the centers, we con- 
structed the triangle J O P (Fig. 272), which we said if revolved upon J O as an axis would generate a cone, a 





Fig. 273. — The Plan shown in the Pre- 
ceding Figure, with a Portion of the 
Solid, of which the Frustum K L N M 
is a Part, in Position. 



The Art and Science of Pattern. Cutting. 



79 





part of which would correspond to a portion of the frustum in question. In Fig. 273, the plan ACBD cor- 
responds to the plan represented by the same letters in Fig. 272. The triangle FDE corresponds to J O P of 
the preceding figure, while the portion of the solid represented in position 
on the plan is a part of the cone, as already explained, that would he 
generated hy the revolution of this triangle about its altitude as an axis. 

456. Fig. 274 shows the same plan, with portions of the small cone, 
corresponding to the end sections of the plan, in position. The triangles 
L F G and M H K correspond with the triangle Z II P of Fig. 272, and 
the small cones are the same as would he generated by the revolution of 
ZEP about its altitude Z R as an axis. As already remarked, we have, 
for the purpose of simplifying the operation, employed but two sets of 
centers in this illustration. From what has preceded, it is evident that if 
more centers were used the solid of which the frustum is a section would 
be composed of parts of a larger number of cones, the joining together 
of which would be upon the same general principle as here explained. 

457. We will now return to the string or thread method, which we 
employed with the right cone, showing how it may be applied to this 

compound solid. A description of 
its use will still further explain the 
usual method of describing the 
patterns of regular flaring ware. 
Since all pattern cutting is, in result, 
a system of measurements upon 

the surface of the various solids, envelopes of which are required, occa- 
sional experiments in measuring the solids themselves, instead of always 
dealing with representations of them, are advantageous. Hence our 
experiments with the clay molding, the revolution of solids, and this 
string method of describing the envelopes of cones. Fig. 275 shows the 
opposite side of the parts presented in Fig. 274, with a pin fastened 
at the apex, and a thread 
attached carrying points G 
and H, representing the two 
surfaces of the frustum. ISTow, 
if we draw the string tight, 
and pass it along the side of 
the larger segment of the 
cone from A to B, the points 
will follow the upper and 
lower bases of the frustum. 
When we reach the point B, 
if the finger be placed upon 



B K 

Fig. 274. — The same Plan, with Portions 
of the Small Cone shown in Position, 
Joined to the Larger One. 



Fig. 275. — The Opposite Side of the Parts 
in Fig. 274, showing a String attached to 
a Pin fastened at the Apex of the Larger 
Segment. 

the thread at the apex of the lesser cone, as shown by C, and the 

progress of the thread be continued, the points will still follow the 

lines of the bases of the frustum. If the pin and thread be taken 

from the cone and transferred to a sheet of paper, as shown in Fig. 

276, the pin A being used as a center and the thread as a radius, 

the points will describe the envelope of the frustum. First, the 

radius is used full length, as shown by A L K, and arcs L M and 

K II are drawn, in length equal to the base of the larger segment 

of cone in the solid, Fig. 275. Then a second pin is put through 

the string, as shown at B, thus reducing the radius to the length 

of the side of the lesser cone, and arcs are struck in continuation of those first described, making the length of 

the additional arc equal to the base of the segment of the small cone. 




Fig. 276. — The Pin and Thread taken from 
the Solid and Employed in Describing the 
Envelope. 



80 



The Art and Science of Pattern Cutting. 



458. In our description of the solid of which the frustum that has equal flare all around is a part, we have 
called it a compound shape. In Figs. 273 and 274 we have shown parts of the cones corresponding to the tri- 
angles constructed in Fig. 272, which compose it. The larger cone employed has such diameter of base as 
causes its axis to fall upon the opposite boundary line of the plan, as shown by D in Fig. 273 and B in Fig. 
274. Now, if it were desired to complete the solid — that is, to employ other portions of the cone to fill up the 
blank spaces in the plan— it would first be necessary to reduce the larger segment by cutting it upon a line cor- 
responding to C D of the plan in Fig. 274. This line would pass through the top in the points L and M. By the 
nature of the shape with which we have to deal, the shape of the top of the solid thus cut would be a hyperbola. 
(See Section 120.) Completing the solid as above suggested, by adding a second section of the large cone, would 
produce the form shown in Fig. 277. To look at this solid, or to look at an ordinary elliptical flaring pan, affords 

little or no suggestions as to the possible composition of the shape and the 
rules for cutting the patterns which are to be deduced therefrom. Yet it 
is by such analyses as we have above described that the science of pattern 
cutting is to be understood. 

459. We might extend this chapter, entering still further into the rea- 
sons and methods of sheet-metal pattern cutting, 'but enough has been writ- 
ten to afford the intelligent student such an insight as will enable him to 
continue investigations in other directions for himself. So we shall bring 
our talk about the art and science of the subject to a close at this point, add- 
ing only a few words of general advice. "We would caution the student 
against arbitrary rules and methods. We think we have demonstrated con- 
clusively that there are governing principles underlying all operations what- 
soever. Therefore we say, search for the reason of every step to be taken. 
Do not be content to follow a rule because it is a rule. There should be no rules in pattern cutting, using that 
word in the sense in which it is ordinarily employed. There are principles and the application of principles, 
but not set rules. The good sense of the student must govern him in the employment of principles and in the 
choice of methods. There is hardly a pattern to be cut which cannot be obtained in more than a single way. 
Under some conditions one method is best, and under other circumstances another. Careful thought before the 
drawing is commenced will show which is best for the purpose in hand. To make this book of the greatest 
possible usefulness, we have added quite an extensive list of problems and demonstrations, but the methods we 
have employed are not to be taken as fixed rules. The same results in almost all cases may be reached by differ- 
ent methods. The student, therefore, should learn to choose between the different ways open before him and 
to work intelligently, otherwise he will not attain the highest degree of proficiency in his art. 




Fig. 277. — The Solid of which a Reg- 
ular Flaring Elliptical Frustum is 
a Part. 



81 



PATTERN PROBLEMS. 




460. Having in the preceding chapters defined the terms most frequently employed in pattern cutting, 
shown how drawing tools and materials may be employed most advantageously, explained the geometrical prob- 
lems of most common occurrence in practical work and the 
general theory of pattern cutting, we will now complete our 
task by presenting a selection of pattern problems, so arranged 
as to be convenient for reference upon the part of those who 
make use of this portion of the book without previous study 
of the other chapters. "We shall attempt, therefore, to make 
each demonstration complete in itself and to avoid refer- 
ences to other parts of the book. To do this, we must assume 
for the reader a certain degree of familiarity with general 
principles and methods. If any one fails to comprehend any 
of the steps described, we suggest that his difficulties may be 
overcome by turning to those parts of the book where elemen- 
tary matters are explained. 

461. The Envelope of a Triangular Pyramid. — Let ABC 
of Fig. 278 be the elevation of the pyramid, and E F G of Fig. 
279 the plan. Draw the lines E E, F Iv and G K in the plan, 
representing the angles. From the end of any one of them, as 
K of the line F K, erect a perpendicular, as K II, equal in length FJ e- 2i ■- pa " ern - 

to the hight of the pyramid, as shown by A D of the elevation. ™° ^«**» °f a Triangular Pyramid. 

Draw F H, which then represents the length of the corner lines. From any point, as L of Fig. 280, for center, 

with radius equal to F H, describe the 
arc M X O I indefinitely. Draw L M. 
From M set off the chord M N, in 
length equal to the side F G of the 
plan. In like manner set off N O and 
O I respectively, equal to G E and 
E F of the plan. Connect I and L, as 
shown, and draw L O and L jST. Then 
L I O 1ST M is the pattern sought. 

462. The Envelope of a Square 
Pyramid.— Let E A C of Fig. 281 be 
the elevation of the pyramid, and 
F II K L of Fig. 282 the plan. The 
diagonal lines F K and LH of the plan 
represent the angles or corners, and G, 
a point corresponding to the apex A of 

The Envelcpe of a Square Pyramid. ^ elevation _ From the apex A drop 

the line A B perpendicular to the base E C. Prolong E C in the direction of D, making B D equal to G F of 




Fig. 



Fig. 283.— Pattern. 



82 



Pattern Problems. 



the plan. Connect D and A. Then A D will be the slant hight of the article on one of the corners, and the 
radius of an arc which will contain the pattern, as shown in the diagram. From any center, as M, Fig. 283, 
with a radius equal to A D, describe an arc, as P R O S N, indefinitely. Draw M P. From P set off a chord, 
P R, in length equal to one of the sides of the pyramid shown in the plan. From R set off another chord, 
R O, in like manner, and repeat the same operation for O S and S X. Draw M X, and likewise MS, MO 
and M R Then MNSOEP will be the required pattern. 

463. The Envelope of the Frustum of a Square Pyramid. — In Fig. 284, let G II K I be the elevation of 
the article, C A E D the plan of the larger end and LMON the plan of the smaller end. Produce the miter 

lines C L, A M, etc., in the plan to the center P. Construct a diagonal 
section on the line A P as follows : Erect the perpendicular P F, mak- 
ing it equal to the straight hight of the article, as shown by R K of 
the elevation. Likewise erect the perpendicular M B of the same 
length. Draw F B and A B. Then P A B F is the diagonal section 
of the article cut on the line P A. Produce A B indefinitely in the 
direction of X, and also produce P F until it meets A B extended in 
the point X. Then X is the apex of a right cone and X B a side of the 
same, the base of which, if drawn, would circumscribe the plan C A E D. 
Therefore, from any convenient center, as X 1 of Fig. 285, with X A 
as radius, describe the arc C 1 D' E 1 A' C, and from the same center, 
with radius X B, draw the arc L' X' O' M' L 2 , both indefinitely. Draw 
C L". Make the chord C D 1 equal to one side, C D, of the plan, and 
D 2 E 1 to another side, D E, of the plan, and so on. Draw D 1 X', 
E' O 1 , etc., which will represent the lines of bend in forming up the 
pattern. Draw the chords L 1 N 1 , X' O 1 , etc., thus completing the 
pattern. 

4G4. The Envelope of a Hexagonal Pyramid. — Let H G I of Fig. 
286 represent the elevation of a hexagonal pyramid, of which D F C L B E 
of Fig. 287 is the plan. The first step is to construct a section on a line 

drawn from the center of the figure 
through one of its angles, as shown 
in the plan by A B. From the cen- 
ter A erect A X perpendicular to 
A B, making it equal to the straight 
hight of the article, as shown in 
the elevation by G K. Draw B X. 
Then X is the apex and X B one 
of the sides of a right cone, the 
plan of the base of which, if drawn, 
would circumscribe the plan of the 
hexagonal pyramid. From any con- 
venient center, as X' of Fig. 288, 
with X B as radius, describe an arc 
indefinitely, as shown by the dotted 
line. Through one extremity to the 
center draw a line, as shown by 
D'X'. "With the dividers set to a space equal to any side of the plan, commencing at D 1 set off this distance on 
the arc six times, as shown. From the several points E' B 1 L 1 in the arc thus obtained, draw lines to the center, 
as shown by E 1 X 1 , B 1 X 1 , etc. These lines represent the angles of the completed shape, and serve to locate the 
bends to be made in process of forming up. 

465. The Envelope of the Frustum of an Octagonal Pyramid. — Fig. 2S9 shows the elevation and Fig. 290 
the plan of the frustum of an octagonal pyramid. The first step in developing the pattern is to construct a diag- 
onal section, the base of which shall correspond to one of the lines drawn from the center of the plan through 
one of the angles of the figure, as shown by G B. Erect the perpendicular G C equal to the straight hight of the 




Fifj. 384.— Plun and Elevation. 




Fig. 38;.— Pattern. 
The Envelope of the Frustum of a Square Pyramid. 



Pattern Problems. 



83 




frustum, as shown by N M of the elevation, and at b erect a perpendicular, b A, of like length. 
A C. Then G B A C is a section of the article as it would appear if cut on the line G B 
indefinitely in the direction of X, and likewise prolong G C until it 
intersects B A produced in X. Then X is the apex and X B the side 
of a right cone, the plan of which, if drawn, would circumscribe the base 
of the frustum. From any convenient center, as X 1 , Fig. 291, with radius 
X B, describe an arc indefinitely, as shown by the dotted lines E 1 E 2 of 
the pattern, and from the same center, with X A for radius, describe 
the arc e' e* of the pattern. Through one extremity of each to the 
center draw a straight line, as shown by E 1 e' X'. Set off on the arc 
E 1 E" spaces equal to the sides of the plan of the base of the article 
H N and connect the points by chords. Thus make 

E 1 P 1 of the pattern equal to E P of the plan, 
and so on. Also from these points in the arc 
draw lines to the center, cutting the arc e' e 2 , 
as shown. Connect the points thus obtained 
in this arc by chords, as shown by e 1 p\ jp' d\ 
d' o\ etc. Then e x E 1 E' e ! will be the pattern 
sought. 

4CG. The Envelope of the Frustum of an 

Octagonal Pyramid having Alternate Lw\g 

and Short Sides.— In Fig. 292, let 

IMBNOPKLbe the plan of 

the article of which G II F E is the 



Draw B A and 
Produce B A 



Fig. 236.— ElevaUon. 











■ — 




1 
\ 


\\ 










\. // 












\^ 1 


e'' 


S N / 








> 

// 

\ // 
\ // 






\^a. 


^= 


^C^~^ 





Fig. aS8.— Pattern. 
The Envelope of a Hexagonal Pyramid. 

elevation. The first thing to do in describing the pat- 
tern is to construct a section corresponding to a line 
drawn from the center to one of the angles in the plan, 
as SB. At S erect the perpendicular S K, in length 
equal to the straight bight of the article, as shown by 
CD of the elevation. Upon the point b erect a cor- 
responding perpendicular, as shown by b A. Draw 
E A and A B. Then B A K S is a section of the article 
taken upon the line S B. Produce S R and B A until 
they meet in the point X. Then X is the apex and 
X B is the side of a cone, the base of which, if drawn, 
would circumscribe the plan of the article. From any 
convenient center, as X', Fig. 293, with radius equal to 
X B, describe an arc, as shown by M 1 W. Draw X 2 M* as one side of the pattern. Then, starting from M 1 , set off 
chords to the arc, as shown by M 1 B\ B 1 W, etc., equal to and corresponding with the several sides of the article, as 
shown by M B, B N, etc., in the plan. From these points, B 1 , N", etc., in the arc, draw lines to the center X'. 



o' 

Fig. 291- 
The Envelope of the Frustum of an Octagonal Pyramid. 



84 



Pattern Problems. 



with X A 



radius, describe an arc cutting these lines, as shown by m 1 m'. Connect the points of inter- 
section by straight lines, as shown by to' J 1 , 1' n\ n' o\ etc. Then 
m' ?ri J M 2 M 1 will be the pattern sought, and the lines B 1 b\ N" 1 n\ 
etc., will represent the lines of bends to be made in forming up 
the article. 

467. The Envelope of the Frustum of a Pyramid which is 
Diamond Shape in Plan. — In Fig. 294, let A B D E be the 
elevation and K G I O the plan of the pyramid at the 
base. Complete the plan by drawing MLPE, the shape of the 
top of the article, and draw the angle lines 6 L, I P, O E and 




Fig. 392. —Plan and Elevation. Fig. 293.— Pattern. 

The Envelope of the Frustum of an Octagonal Pyramid having Alternate Long and Short Sides. 

K M. Draw C F in the elevation, representing the angle G L of 
the plan. It also serves to measure the straight bight of the 
frustum. At right angles to M E of the plan draw S "W", making 
its length equal to the straight bight of the frustum, as shown by 
C F of the elevation. Through W draw N H indefinitely, par- 
allel to K O. At right angles to K O, through the points K and O, 
draw lines, K N and O H, cutting 1ST H in the points N and II, 
thus establishing its length. Connect M N and Ii II. Then 
MEM will be the pattern of one of the four sides composing 
the article. 

46S. The Pattern of a Rectangular Flaring Article. — 
In Fig. 295, let C A B E be the side elevation of the article, 
of which FIKM is the plan at the base and GHLN the 
plan at the top. Let it be required to produce the pattern in 
one piece, the top included. Make H 1 L 1 N 1 G' in all respects 
equal to H L N" G of the plan. Through the center of it length- 
wise draw B P indefinitely, and through the center in the oppo- 
site direction draw O S indefinitely. From the lines IT L 1 and 
G 1 N 1 set off T O and W S respectively, each in length equal to 
the slant bight of the article, as shown by C A or E B of the 
elevation. Through O and S respectively draw I" K 1 and F 1 M', 
parallel to H 1 L 1 and G 1 W, and in length equal to the correspond- 
ing sides in the plan I K and F M, letting the points O and S fall midway of these lengths respectively, 



Elevation 




Pattern 



Fig. 294. — The Envelope of the Frustum 
mid which is Diamond Shape in 



of a Pyra- 
Plan. 

as shown. 



Pattern Problems, 
In like manner set off V P and U E, and draw tin- 



85 



gh E and P the lines P P and K= W, parallel to the 
ends ot the pattern of the top part as already drawn, and in length equal to I F and K M of the nlan Draw 



P IP, K' L 1 , E? L', M 2 N 1 , M 1 W, F' G 1 , ]«" G 1 and 
I 2 H 1 , thus completing the pattern sous 



ht. In the 

same general way the pattern may be described, 
including the bottom instead of the top, if it be 
required that way, or the sides may be developed 
independent of either top or bottom. 

469. The Pattern of a Rectangular Article, 
Three Sides of which are Vertical, the Fourth leing 
Inclined. — In Fig. 296, let L H I K be the eleva- 
tion and CBAFED the plan. Let it be required 
to describe the pattern in one piece, locating the 
seam at the point G in the plan. Draw L 1 G 1 indefi- 
nitely. From G 1 , at right angles to L 1 G 1 , draw 
G 1 0, in length equal to H I, the straight hight of 
the article in the elevation. Draw O E 1 indefinitely, 
parallel to G l IA From G 1 , which represents the 
end of the pattern, set off G' IP, equal to G A of 
the plan. Draw IF F 1 at right angles to L 1 G 1 , 
cutting O E" in F 1 . From IP set off IF L 1 , in length 
equal to II L of the elevation, and from F 1 set off 
F' E', equal to I K of the elevation. Draw L 1 E 1 , 
which corresponds to L K of the elevation. At 
right angles to L 1 E', from L 1 , draw L 1 D 1 , in length 
equal to the width of the article, as shown by C D 
of the plan. In like manner draw E 1 K' of the 
same length. Draw D 1 K 
erect the perpendicular E 1 M. 



Elevation. 



Upon the point E 1 
With the dividers 




Elevation 



Fig. 295. — The Pattern of a Rectangular Flaring Article. 

set to the distance M L', from D 1 as center, strike the 



arc a I, and with the radius E' M, from K' as center, strike 
the intersecting arc y x. Then the point of intersection, or 
M 1 , corresponds to M of the other arm of the pattern, and 
is a point through which the line of the side must pass. 
Therefore, from D 1 through M' draw D 1 E, and parallel to 
it, from K', draw K' S. Make these lines respectively 
equal to II L and I K of the elevation and connect their 
extremities, which will complete the pattern. 

ito. The Pattern of a Rectcmgular Flaring Article 
having One End Upright.— Lei A B G II of Fig. 297 be 
the side elevation of the required article, and CUVD the 
elevation of the end, or section — both being the same in 
this case. Construct a plan, as shown by L M T S, Fig. 298, 
making L M and S T equal to A B of the elevation, and 
L S and M T equal to C D of the profile. Also make N P 
and O E equal to U V, and 1ST O and P E equal to H G. 
From these three views of the article the pattern may be 
obtained as follows : Lay off W 0' E 1 P 1 , Fig. 299, equal 
to N O E P of the plan, and through the center of it draw 
Make G 1 B' equal to G B of the elevation, and through B 1 , parallel to O' E', draw C D 1 , in 

Draw C O 1 and D" E'. Produce 




Fig. 296. — The Pattern of a Rectangular Article, Three 
Sides of which are Vertical, Uie Fourth being Inclined. 



V B 1 , as shown. 

length equal to C D of the profile, placing one-half on each side from B 1 



86 



Pattern Problems. 



making O 1 M 1 and E 



O 1 E 1 in the direction of M 1 and T 

T 1 , parallel to the plan of the bottom already drawn 

by 



Side and End Elevation. 




T" each equal to C U of the profile. From M 1 and 
, draw M 1 L 1 and T 1 S 1 , in length equal to the sides, as 
shown by M L and T S in the plan. Connect L 1 1ST 1 and S 1 P'. 
Make II 1 A 1 equal to II A of the elevation. Through A 1 draw 
S 2 IA parallel to P 1 ~N l , and in length equal to C D of the profile, 
placing equal portions of it each side of A 1 . Connect L" 1ST 1 and 
S 2 P 1 , thus completing the pattern. It is to be observed that the 
plan M T S L is not absolutely necessary in describing the pattern. 
All the necessary measurements may be obtained from the profile 
and the elevation. It is given in this connection, as more clearly 
showing the principles pertaining to patterns of this class than could 
be done without it. 



Fig. 298.— Plan. 




M 1 



g l 4 



Fig. 299. — Pattern. 

The Pattern of a Rectangular Flaring Article 
having One End Upright. 

471. The Pattern of a Flaring Article of 
which the Base is an Oblong and the Top Square. 
—Let A B D E of Fig. 300 be the elevation of 
the article, and F N O I the plan. Let KMPL 
represent the top of the article. If the article 
is to be used as a cover, the top being solid and 
the bottom open, proceed for the pattern as fol- 
lows : Draw K 1 M 1 P' L 1 , Fig. 301, equal in all 
respects to K M P L of the plan. Through the 
center of it, and at right angles to each other, 
draw lines Y U and S T indefinitely. Through 
the elevation, and perpendicular to the base and 
top, draw the line C G, which will measure 
the straight bight of the article. From G 
set off G H, in length equal to M E of the 
plan. Draw H C. Then H C will be the slant 



bight 



of the article on the side, and therefore 




the width of the pattern of that portion. The 
slant bight of the article at the ends, or the width 
of the pattern for the ends, is shown by A B 
and E D of the elevation. Upon V U of the 
pattern, from K 1 M 1 set off W Y, and from L 1 P 1 
set off X TJ, in length equal to H C of the eleva- 
tion, and upon S T set off Z T from M 1 P 1 , and Fig. 3 oi.-Pattem. 

T S from K 1 L 1 , in length equal to A B or D E The Pattern of a Flaring Article of which the Base is Oblong and the 

of the elevation. Through TJ and V draw lines Top Square - 

parallel to K 1 M 1 and L 1 P 1 , making them in length equal to F N and I O of the plan, letting the points V 
and TJ come midway of their lengths respectively. Draw F 2 Iv 1 , N 1 M 1 and I 1 L 1 , O 1 P 1 . In like manner through 
the points S and T draw F 2 P and W O 2 parallel to K 1 L 1 and M 1 P 1 , and in length equal to F I and 1ST O of the 



Pattern Problems. 



87 



plan, letting the points S and T fall midway of their lengths respectively. Draw F 2 K', P L 1 and W M' 5 P' 
which will complete the pattern. If the pattern is wanted in four pieces instead of one, as above described' 
set off K 1 M 1 , upon which erect K' F 1 W W in the same manner as explained, and likewise upon M 1 P 1 erect 
M ! K 2 O 2 P 1 . The other lines and parts may be dispensed with. 

472. The Patterns of a Tapering Article which is Square at the Base and Octagonal at the Top. 

ABDC in Fig. 302 shows the plan of the article at the base, IKLHHGFE represents the shape 
at the top, E 3 IF D 2 C 3 is an elevation of one side. Con- 
struct a diagonal elevation, as shown by I 1 G 1 D 1 A 1 , as fol- 
lows : Extend the base line D 2 C 3 of the elevation, as shown, 
making D 1 A 1 equal to the diagonal length across the plan, 
as shown by D A. In like manner extend the top line IP E 3 
of the elevation, making G 1 I 1 equal to the distance from G 
to I of the plan, letting the middle point Ft 1 in I 1 G 1 fall 
directly above the middle point C 2 iu A' D 1 . Draw I 1 A 1 
and G 1 D 1 . Then I' G 1 D 1 A 1 is an elevation or section of 
the article taken upon the line A D of the plan, and there- 
fore A 1 I 1 represents the length of one of the smaller sides 
of the article. Produce the diagonal line R C, as shown, 
making N C 1 in length equal to I 1 A 1 of the diagonal sec- 
tion. By means of the T-square, as indicated by the dotted 
lines, set off E 1 F 1 equal to E F of the plan and draw C 1 E 1 
and C F. Then E 1 C 1 F 1 is the pattern 
of one of the smaller sides of the article. 
From the center R of the plan draw R P 
perpendicular to the side A C, upon 
which set off O P, in length equal to 
E 3 C 3 of the elevation. At right angles 
to it draw A 2 C, which, by means of the 
T-square, as shown by the dotted lines, 
make equal to A of the plan. In like 
manner draw I" E' equal to I E of the 
plan. Connect A 3 P and C 4 E 4 . Then 
A 3 P E 4 C 4 will be the pattern of one of 
the larger sides of the article. If for any 
reason the pattern is desired to be all in 
one piece, the shapes of the different sides 
may be laid off adjacent to each other, 
the large and small sides alternating, all 
as indicated by i i' a a\ Fig. 303. 

473. The Pattern of a Flaring 
Article Square at the Base and Bound 
at the Top.— Let P R T S of Fig. 
304 be the elevation of the article, 
GBKH, Fig. 305, the plan at the base, 
and L 1ST M the plan at the top. The 
corners, one of which is shown at 
M K N, are to be regarded as sections 
of oblique cones, the apexes of which 
lie in the angles of the plan of the 
base, or, in the case above cited, in the 
point K. The first step in developing the pattern is to construct a diagonal section, by which to get points 
from which to describe the envelope of that portion of the cone forming the corners. At any conveni- 
ent place draw A 1 B 1 parallel to A B, and in length equal to it, which may be established by means of the 




D' C 5 

Fig. 302.— Elevation and Plan and Patterns. 




Fig. 303. —Pattern in One Piece. 
The Patterns of a Tapering Article which is Square at the Base and Octagonal at 

the Top. 



88 



Pattern Problems. 



T-square, as indicated by the dotted lines A A 1 and B B 1 . From A 1 , on A 1 A, set off A 1 A', in length equal to 
the straight hight of the article, as indicated by U V of the elevation. From A 2 draw A 2 C parallel and equal 
P u R to A C of the plan. Draw A 2 B 1 and C B 1 . Then A 1 A 3 C B 1 is a diagonal sec- 

tion of the article corresponding to the line A B of the plan, and A 3 B 1 C repre- 
sents a section of the oblique cone form- 



ing the corners 



Fig. 304.— Elevation. 



B 1 is the aj)ex, B 1 A 2 is 
the axis, and B 1 C is one of the sides, 
while NCOof the plan is a section of 
its base. Divide NCO into any con- 
venient number of equal parts, and from 
the points thus obtained drop points per- 
pendicular to A B on to A 3 C\ 
From A 2 C carry the points in 
a direction perpendicular to the 
axis A 2 B' until they meet B 1 C, 




Fig. 306.— Pattern. 
The Pattern of a Flaring Article Square at the Base and Round at 

the Top. 

extended as shown by C D. From B 1 as center, strike arcs corre- 
sponding to each of the several points in C 1 D, as shown. From 
any convenient point in the first arc, as E, draw a line to B 1 , as 
shown by E B 1 . Set the dividers to the space used in stepping off 
the plan KCO and, commencing with the point E", lay off the 
stretchout of N C O, stepping from arc to arc as shown, the last 
point being F. Draw F B 1 , and trace a line through the points in 
the arcs, as indicated by E F. Then E B 1 F is the pattern of one 
of the corners. For convenience in laying off the pattern in one 
piece, transfer this part of the pattern to any space sufficiently 
removed from the diagram of the plan to avoid confusion of lines, 
as shown by E 1 B 2 F 1 in Fig. 306. To this add the triangle forming one of the sides, in the following manner : 
From E 1 as center, with E' B 2 as radius, describe an arc, as shown by B 2 K 1 . From B 2 as center, witli radius 




Fig 308.— Pattern. ^M 
The Pattern of a Regular Flaring Oblong Article 
with Round Corners. 



Pattern Problems. gg 

B K of the plan, intersect that arc in the point K\ Draw B< K' and K 1 E\ To this in turn add the shape of 

n° 6 T ■ M' G ?° merS ' a f d ?° n f n ™ the °P eration xmtil tlie entire number of sides are represented, all as shown by 
O 1 L 1 M' E 1 F 1 B J E? H' G' B 3 , which shows the required pattern complete in one piece. 

474. The Pattern of a Regular Flaring Oblong Article with Pound Corners.— In Fig. 307 A C D B is 
the side elevation of the article and EFGMOPK the plan. 
The corners are arcs of circles, being struck by centers H L T S, as 
shown. Draw the plan in line with the elevation so that the same 
parts in the different views shall correspond. Through the centers 
H and L of the plan by which the corners F G and M N are struck, 
draw FN indefinitely. Prolong the side line of the elevation C D 
until it cuts F N in the point K, as shown. Then K D is the 
radius of the inside line of the pattern of the curved part, and K C 
is the radius of the outside line. Draw the straight line E 1 F 1 of 
Fig. 308 in length equal to the straight part of one side of the 
article, or E F of the plan. Through the points E" and F 1 , at right 
angles to the line E 1 F 1 , draw lines indefinitely, as shown by E' U 
and F 1 K 1 . Set the compasses to the radius K C, and putting the 
pencil point at F 1 , establish the center K' in the line F 1 K 1 . Strike 
the arc F 1 G 1 , which in length make equal to F G of the plan. From 
G 1 draw a line to the center K 1 , at right angles to which erect G' M', 
in length equal to G M of the plan. In like manner establish the 
center IT in the line E 1 IT, and from it, with like radius, describe the 
arc E' E 1 . Draw E 1 IT, at right angles to which erect E 1 P 1 , equal 
to E P of the plan. At right angles to E 1 P 1 draw P 1 V indefinitely. 
In the manner above described establish the center V, and from it 
describe the third arc P 1 O 1 . Draw O 1 V. At right angles to it 
lay off O 1 W, equal to O N of the elevation. Draw N 1 W, and 
draw the arc N 1 M 2 in the same manner as already described. In 
the same manner lay off the inner line of the pattern, as shown by 
m g f e r p o n ml. Join the ends M 1 m and M 3 m 1 , tlms complet- 
ing the pattern sought. 

475. The Pattern of a Regular Flaring Article which in 
Shape -is Oblong with Semicircular Ends. — In Fig. 309, let 
A B D C be the side elevation of the required article. Below 
it and in line with it draw a plan, as shown by E c d F H G. From 
D in the elevation erect the perpendicular D L. Then L C repre- 
sents the flare of the article and C D is the width of the pattern 
throughout. Across the plan at the point where the curved end 
joins the straight sides, draw the line d II at right angles to the sides 
of the article. As the plan may be drawn at any distance from the 
the elevation, this line must be prolonged, if necessary, to meet 
C D extended. Produce C D until it meets d II, as shown by g. 
Then g D and g C are radii of the curved parts of the pattern. 
Lay off on a straght line, M O in Fig. 310, the length of the 
straight part of the article, as shown in the plan by c d. At right 
angles to M O draw M S and O E indefinitely. , On these lines set 
off the spaces M E" and O P respectively, both in length equal to 
C D, the slant bight of the article, which must be the width of the 
pattern. Set the compasses to the space g C for radius, and putting 
the pencil to M, establish the center S, which must fall somewhere 
in the line M S. From this center strike the arc M IT indefinitely. 
In like manner, with same radius, describe the arc O V. From the 
same centers, with radius equal to g D, describe the arcs N T and P W. 





Fig. 31c— Pattern. 

Tlie Pattern of a Regular Flaring Article which 
in Shape is Oblong with Semicircular Ends. 

Step off the length of the curved 



90 



Pattern Problems. 



part of the article upon either the inner or outer line of the plan, and make the corresponding arc of the pat- 
tern equal to it, as shown by the spaces in 1ST T and P W. Through the points T and W draw lines from the 

centers S and R, producing them until they cut the outer arcs at U 
and V. At right angles to the line S T U or It W V, as the case may he, 
set off V X Y W, equal to M O P N, which will he the other straight 
side of the pattern. Then UMOVXYWPNT will be the com- 
plete pattern in one piece. If it were desired to locate the seam mid- 
way of one of the straight sections, in adding the last member, as above 
described, one-half would be placed at each end, instead of all at one 
end, as we have shown. In like manner changes may be made locating 
the seam at other points, or for cutting the pattern into several pieces. 
476. — The Pattern of a Raised Cover, Fitting an Oblong Vessel 
with Pound Ends. — In Fig. 311, let A B C D represent a side ele- 
vation of the cover of which E G F II is the plan or shape of the ves- 
sel it is to fit. Various constructions may be employed in making 
such a cover as this ; that is, the joints, at the option of the mechanic, 
may be placed at other points than shown here ; the principle used in 
obtaining the shape, however, is the same, whatever may be the location 
of the joints. By inspection of the elevation and plan it will be seen 
that the shape consists of the two halves of the envelope of a right 
cone, joined by a straight piece. Therefore, for the pattern we pro- 
ceed as follows : At any convenient point lay off B 2 C J , in length 
and C 2 




M 



Fig, 



M K L 

Elevation Plan and Pattern. 
311- — The Pattern of a Raised Cover, 



Fitting an Oblong Vessel with Round Ends. 



equal to B 1 C of the plan. From B 
as centers, with radius equal to A B or C D 
of the elevation, describe arcs, as shown by 
O N and P M. Upon these arcs, measured from 
O and P respectively, set off the stretchout of 
the semicircular ends, as shown in plan, thus 
obtaining the points M and N. From 1ST draw 
N B 2 , and from M draw M C\ From B 2 and C 2 , 



Pattern of 8id« 



at right angles to the line 



B 2 C 2 , draw B 2 K and 



C 2 L, in length equal to A B of the elevation, 
which represents the slant hight of the article. 
Connect K and L, as shown. Then ONKIMP 
will be the required pattern. 

477. The Patterns of a Flaring Article 
Oblong in Plan with Rounded Corners, and 
having Or eater Flare at the Ends than at the 
Sides. -In Fig. 312, let A B D C be the eleva- 
tion of the article and E G II F the plan. 
A K C of the elevation represents the flare of 
the ends, while LNI represents the flare of 
the sides. We will describe the pattern of the 
sides and ends, the latter including the rounded 
corners, as cut separately, although the two may 
be joined in one piece, or the entire rim may be 
constructed of one piece if required. For the 
pattern of the side, as indicated by P S R in 
the plan, draw R 1 S 1 parallel to the side of the 
plan, and in length equal to R S. Draw a per- 
pendicular to it, M 1 K 1 , in length equal to L M. 
Through N 1 draw O 1 P 1 , also parallel to the side 
of the plan and equal to O P. 




Fig. 312. — The Patterns of a Flaring Article Oblong in Plan with 
Rounded Corners, and having Greater Flare at the Ends than at 
the Sides. 



The length of O 1 P 1 and R 1 S 1 may be readily determined by using the T-square, 



Pattern Problems. 



91 






/ 



/ 



/ 



I 



as indicated by the dotted lines. Draw 1 R' and P 1 S 1 . Then 0' P' S' R' will be the pattern of the side. In 
obtaining the pattern for the corners they must be considered as parts of cones. An elevation of the section of 
the cone rrmst be constructed with its 
base on a line parallel to a line drawn 
through the centers by which the curves 
in the plan were struck. It is a matter 
of convenience to draw the elevation of 
the cone in connection with the elevation 
of the article, as shown. Through the 
centers a b, by which one of the corners 
of the plan is struck, draw af. Use the 
top line A B of the elevation, which is 
parallel to a f for the base of the cone. 
Prom a drop a perpendicular to the base 
of the elevation, thus establishing the 
point a\ In like manner drop a line 
from b to the upper line of the elevation, 
establishing the point V. A correspond- 
ing point toy is B, and to d is D. Draw 
V a 1 , producing it indefinitely in the 
direction of x. Also produce B D until 
it meets V a 1 in the point x. Then x 
will be the apex of the cone, a section 
of which constitutes the corner of the 
article. Divide the plan of the cone ef 
into any convenient number of equal 
parts, as shown by the small figures. 
From each of these points drop a per- 




. 313.— Elevation and Plan. 
A\- R ' " 



AMD* 

X 

Fig. 314.— Diagram 
of Small Cone. 



Fig. 316— Diagram 
of Large Cone. 




Fig. 315.— Diagram 

of Middle Cone. 



An Oval or Egg-Shaped Flaring Pan. (For Pattern see Next Page.) 



pendicular to the base b' B of the cone, 

as shown, and from the points in it thus determined carry lines toward the apex 

sider x b l as .the axis of the cone. Therefore from the points in b' B and a 1 D, 

points to D B, as shown. 

described, all as shown. 

pattern. 



cutting the line a' D. Con- 



at right angles to x b\ carry 
From x as center, strike an arc corresponding to each of the points in B D, just 
From x draw any straight line, as x e\ crossing these arcs, which shall be one end of the 
From this line, at the point of intersection with the arc corresponding to point 1 in the plan, set off the 
space of one of the divisions of the plan, stepping to the second arc. From this point set off a corresponding space, 
stepping to the third arc, and so on for each of the spaces set off in the plan. A line traced through these 
points, as shown by e'f\ will represent one side of the curved part. From each of the points in e'f 1 draw lines 
toward the center x, cutting the lower set of arcs. Through the points of intersection thus obtained trace the 
line d d 1 , which will form the other boundary of one of the curved parts. At right angles to/' d l draw/'/ 2 , 
equal toff 3 of the plan, and d' d% equal to d d 3 of the plan. Draw/ 2 d", from which set off a second curved 
section, as shown by/ 2 e' d d% in all respects corresponding to/ 1 e 1 d d\ thus completing the pattern of the 
ends of the article. 

478. The Pattern of an Oval or Egg-Shaped Flaring Pan.— Let A B C D in Fig. 313 represent the ele- 
vation of the article, of which A" K L B 1 M I is the plan. The plan is constructed by means of the centers O, 
P, F and F 1 , as indicated. The patterns, therefore, are struck by radii obtained from sections of the several 
cones of which the article is composed. At any convenient place draw the line P 2 P 1 , Fig. 314, indefinitely, 
which let correspond to P of the plan, and upon it construct a section of the article as it would appear if cut on 
the line A 1 P of the plan. Therefore set off, at right angles to it, A 2 P 2 equal to A' P. Make P 2 D 2 equal to 
the straight hight of the article, as shown by R D of the elevation. Make D 2 A 3 of the diagram equal to D' P of 
the plan. Draw A 2 A 3 , which will correspond to A D of the elevation. Prolong A 2 A 3 
the point P. Then P 1 
plan, and P 1 A 3 is the 



until it meets P 2 P' in 
A 2 is the radius of the outside line of the pattern of the portion indicated by K I of J.he 
radius of the line inside of the same part. In like manner draw the line 0' O 2 , 



, Fig. 



315, corresponding to O of the plan, and construct a section taken on the line B', as shown by O 2 B 2 C 2 C. 



92 



Pattern Problems. 



Pic 



-Mi 



Produce B 3 0' until it meets 3 O l in the point O 1 . Then O 1 C 3 and O 1 B 3 are the radii of the pattern of that 
portion of the article contained "between L and M of the plan. Draw the line F 3 F 3 , Fig. 316, which shall 
correspond to F or F 1 of the plan. Make F 3 E equal to the straight hight of the article, and lay off F 3 L 3 at 
right angles to it, equal to F 1 L of the plan, and E F equal to F 1 Z 3 of the plan. Draw L 3 Z 3 , which produce 
until it meets F 3 F 3 in the point F 3 . Then F 3 l" and F 3 L 3 , respectively, are the radii of the pattern of those 
parts shown by K L and I M of the plan. To lay off the pattern after the several radii are obtained, as described 
above, draw any straight line, in length equal to F 3 F 3 , as shown by F 4 K' in Fig. 317, and from F 4 as center, 
with F 3 V and F 3 L 3 , Fig. 316, as radii, strike arcs, as shown by k l V and X 1 L 1 , which in length make equal to the 
corresponding arcs of the plan Iv L Tc I, as shown. Draw L 1 F 4 . Set the compasses to O 1 C 3 , Fig. 315, and, placing 
the pencil at l\ find the center O 3 in the line L 1 F 4 , from which strike the arc V m\ in length equal to I m of the 
plan. In like manner, from the same center, with radius O 1 B 3 strike the arc L 1 M', equal in length to L M of the 
plan. Draw M' O 3 , which produce indefinitely in the direction of F 5 . Set the compasses to F 3 Z 3 , and, placing the 

pencil on to 1 , establish the center F 5 in the line 
M 1 F 5 , and continue the inner line of the pattern, 
as shown by m 1 i\ which in length must equal 
to i of the plan. In like manner, from the same 
center, with radius F 2 L 3 , describe the arc M' I'. 
Draw I 1 F 5 . Set the compasses to P 1 A 3 , and, 
bringing the pencil point to i\ establish the 
center P 3 somewhere in the line I 1 F 5 . Describe 
the arc i 1 Tc', in length equal to i Tc of the plan. 
In like manner, from the same center, with 
the radius P 1 A 3 , describe the arc I 1 K 3 , in 
length equal to I K of the plan. Place the 
straight-edge against the points P 3 and B? and 
draw Iv 3 Tc, thus completing the pattern. From 
inspection it is evident that the pattern might 
have been commenced „at any other point as 
well as at K Tc of the plan, where we have 
located the joint. If the joint is desired upon 
any of the other divisions between the arcs, as 
L Z, M to, or I i, the method of obtaining it 
will be so nearly the same as above narrated 
as not to require special description. If the 
joint is wanted at some point in one of the 
arcs of the plan, as, for example, at X x, draw 
the line X x across the plan, producing it until 
it meets the center by which that arc of the plan is struck. In laying off the pattern, commence with a line 
corresponding to X F 1 , in place of F 4 K 1 , and from it lay off an arc corresponding to the portion of the arc in 
the plan intercepted by X x, as shown by X L Z x. Proceed in other respects the same as above described 
until the line k K 3 is obtained, against which there must be added an arc corresponding to the amount cut from 
the first part of the plan by X x, as above described, or, in other words, equal to x X Tc K of the plan. 

479. The Pattern of a Heart-Shaped Flaring Tray.— Let E C G' F G C of Fig. 318 be the plan of the 
article, and I X O K the elevation. By inspection of the plan it will be seen that eacli half of it consists of 
two arcs, one being struck from D or D 1 as center, and the other from C or C as center, the junction between 
the two arcs being at G and G 1 respectively. From C 1 draw C 1 F, and likewise draw C G 1 . Upon the point D> 
erect the perpendicular D' C. For the radii of pattern construct a diagram, in which show a profile of the article 
upon the lines C 1 G 1 and D 1 C. Draw X P in Fig. 319 in length equal to the straight hight of the article 
Lay off the perpendiculars X U and P S indefinitely. Upon P S, from P, set off P E equal to D 1 C of the 
plan, and on X U, from X, set off X W equal to D 1 c of the plan. In like manner make P S equal to C 1 G 1 of 
the plan, and X U equal to C g of the plan. Connect U S and W K. Produce P X indefinitely in the direc 
tion of Z. Also produce B "W until it meets P X in the point T, and in like manner produce S U until it 
meets P Z in the point Z. Then Z U and Z S are the radii for that portion of the article contained between 




An Oval or Egg-Shaped Flaring Pan. 



Pattern Problems. 



93 



G 1 and F of the plan, and T TV and T R are the radii of that portion shown from G 1 to E of the plan. To lay out 
the pattern after the radii are established, draw any straight line, as Z' G 2 in Fig. 320, in length equal to Z S of the 
diagram. From Z 1 as center, with Z S as radius, describe the arc CV F 1 , in length equal to G 1 F of the plan. In 
like manner, with radius Z U, from the 
same center, describe the arc g\f\ in 
length equal to g f of the elevation. 
Drawy 1 F 1 . Set the compasses to Y It 
for radius. Place the pencil point at G", 
thus establishing the center Y 1 , which 
must fall somewhere in the line Z' G\ 
From Y 1 , with radius as named, describe 
the arc G" E 1 , which in length make 
equal to G 1 E of the plan. In like man- 
ner, from the 6ame center, with radius 
Y TV, describe the arc (f e 1 equal to the 
arc g e of the plan. Draw e 1 E 1 , thus 
completing the required pattern. 

480. The Pattern of a Flaring Ar- 
ticle, the Top of which is Pound and 
the Bottom of -which is Oblong, with 
Semicircidar Ends. — In Fig. 321, O is 
the center by which the plan of the top 
is struck, and P is the center by which 
one of the semicircular ends is described. 
The elevation is placed so as to cor- 
respond with the plan, as shown by the 
lines connecting the two, E A, K B, M D 
and II C. From O erect the perpendic- 
ular o, and from P erect the perpen- 
dicular P p. Prolong the side line C D of 
the elevation indefinitely in the direction 
of X. Through the points/? and o draw/ o, 
which produce until it meets C D pro- 
longed in the point X. Then X is the apex 
of a cone of which that portion of the ar- 
ticle shown by DCjjo in the elevation, 
and by L/ 1 II / = X in the plan, is a sec- 
tion. Then X p will represent the axis of 
the cone. Divide the profile// IT/?" into 
any convenient number of equal parts. 
From each point in it erect a perpendicu- 
lar to j? C, as shown. From the points 
thus obtained in p C carry lines toward 
the apex X, cutting o D, as shown. From 
the points in/ C, and also from those in 
D, draw lines at right angles to the axis X p, cutting the side X C of the cone, as shown. From X as center, 
strike arcs corresponding to each set of points in X C, as indicated. The arcs from the lower set of points are 
to receive the stretchout in the following manner: From X draw any straight line, as X/ 3 , meeting the first 
arc in the point p 3 , which shall be one end of the stretchout. Set the dividers to the space used in stepping off the 
plan, and, commencing at/ 3 , step to the second arc, and from that point to the third arc, and so on, as shown in 
the engraving. A line traced through these points will be the boundary of the lower side of one of the semi- 
circular ends. From each of these points just described draw a line toward the center X, cutting the upper set 
of arcs, as shown. A line traced through these points of intersection will form the upper edge of the pattern 




The Pattern of a Heart-Shaped Flaring Tray. 



94 



Pattern Problems. 



Pattern, 



of the end piece. From the point L', which corresponds to L of the plan, as center, with L 1 p* as radius, 
describe the arcj? 4 R 1 , and from_p 4 as center, with radius equal top 1 R of the plan, intersect it at R 1 , as shown. 
Draw L 1 R 1 . Then L' ~B} p* is the pattern of one of the sides. To L7 R 1 add a duplicate of the end piece 
already obtained, all as shown by L 1 R 1 E R 3 W, and to W R 3 add a duplicate of the side just obtained, as 
shown by N 2 R 3 ^/, thus completing the pattern. 

4S1. The Pattern of a Flaring Article, the Base of which is a Rectangle and the Top of which is Round, 
the Center of the Top being toward One Mid.— -In Fig. 322, let L P N M be the side elevation of the article, of 
which A D C B is the plan at the base and E G H and K is the plan at the top. Draw two diameters through the 
plan of the top parallel to the sides of the article, cutting the top in the points E, G, II and K. Draw the lines 
in the plan A E, A G, D G, D H, etc., and consider the corner pieces E A G, G D H, etc., quarters of inverted 

scalene cones. The first step will be to obtain 
a profile or section of each of these quarter cones. 
From the center F of the plan of the top draw 
the diagonal lines F A and F D, which shall rep- 
resent in plan the diagonal sections to be 
constructed. At any convenient distance outside 
of the plan draw A 2 O 2 , in length equal to A F, 
and parallel to it. From the point O 2 erect the 
perpendicular O 2 F 2 , in length equal to the straight 
hight of the article, as shown by L of the ele- 
vation. From F 2 , perpendicular to O 2 F 2 , set off 
F 2 S 1 equal to F S of the plan. Draw S 1 A 2 and 
A 2 F 2 . Then A 2 O 2 F 2 S 1 is a section of the arti- 
cle taken diagonally from the center F on the 
line F A. Divide one-quarter, G E, of the plan 
of the top, which forms the base of the cone of 
which the corner is a section, into any convenient 
number of equal parts, as shown by the small 
figures 5, 6, 7, etc., and from these points carry 
lines perpendicular to A F, producing them until 
they cut F 2 S 1 , and thence at right angles to 
F 2 A 2 until they cut A 2 S 1 prolonged. Then 
from A 2 as center, describe arcs corresponding to 
these several points, as shown. From A 2 draw 
any straight line, as A 2 E 2 , cutting the first arc 
in E 2 . "With E 2 as a starting point, and with 
the dividers set to the distance used in spacing 
the plan E S G, step to the second arc, and 
thence to the third, and in this manner lay off 
the stretchout, ending in the point G 3 . Trace a 
line through these points in the arcs, as shown, 
and draw G 3 A 2 . Then G 3 A 2 E 2 will be the pat- 
tern of the corner shown by G A E of the plan. 
In the same general manner construct a diagonal 
section corresponding to the line F D of the plan, 
all as shown by O 1 F 1 R D 1 . Divide the quarter-circle G H of the plan into any convenient number of equal 
parts, and from the points thus obtained erect lines perpendicular to F D, continuing them until they cut F' R 1 , 
and thence carry them at right angles to F' D l until they meet D 1 R 1 produced. From D 1 as center, 
describe arcs corresponding to the several points in D 1 R 1 prolonged, upon which, commencing at any point in 
the first arc, as G 1 , step off the stretchout of the plan, stepping from arc to arc in the same general manner as 
explained in connection with the section already constructed. Draw G 1 D' and IF D 1 , and trace a line 
through the points stepped off in the arcs, as shown from G 1 to H 1 . Then G 1 D' IF will represent the pattern 
of the corner shown by H D G of the plan. From II 1 as center, with H 1 D 1 as radius, describe an arc, as indi- 




R 2 G 

Fig. 321. — The Pattern of a Flaring Article, the Top of which is 
Round and the Bottom of which is Oblong, with Semicircular 
Ends. 



Pattern Problems. 



95 



cated by the dotted line, and from D 1 as center, with radius equal to D C of the plan, intersect it in the point 
C. Then D 1 IT C represents the pattern of the end D H C of the plan. To this add a duplicate of IT D" G 1 , 
as shown by H 1 C K\ From C as center, with radius C B of the plan, describe an arc, and from K" as center, 
with radius A 2 E 2 of the pattern for the small corners, describe an arc, cutting it in the point B 1 . Draw the con- 
necting lines. Then C K 1 B' is the pattern of the side shown by C K B of the plan. To K 1 B' add the shape 
of the pattern of the small corner E 2 A 2 G 3 , already obtained, and as shown by B 1 K 1 E'. From E 1 as center, 
with E 1 B 1 as radius, describe an arc, as shown by the dotted line, and from B 1 , with radius equal to B A of the 
plan, intersect it in the point A 1 . Draw the connecting lines. Then E 1 B 1 A 1 is the pattern of the end, as 
shown by E B A of the plan. To this in 
turn add a duplicate of the pattern of the 
small corner, and then a duplicate of the 
pattern of the side, thus completing the 
required shape. 

482. The Pattern of an OUong Flar- 
ing Article having a Pound Top. — In Fig. 
323, A B C D represents the elevation, 
G IT I K the plan of the base, and L M N O 
the plan at the top. Corresponding points 
in the two views are connected by dotted 
lines. In describing the patterns we will 
consider the corners as quarter cones having 
oblique bases, and the sides and ends as flat 
triangles. In other words, the corners are 
sections of a cone, the elevation of which is 
shown by A B E, in which A is the apex 
and B E the oblique base, and the sides are 
simple triangles, as shown by KOI and 
INHof the plan. From G, which repre- 
sents the apex of one of the cones in plan, 
draw a line to the center of the top P, as 
shown by G P, which will represent in the 
plan a diagonal section of the corner, which 
must be constructed to obtain the measure- 
ments necessary. From any convenient 
point outside of the plan, as x, draw x p 
equal and parallel to G P. From p erect 
the perpendicular p a, in length equal to 
the bight of the article, as shown by F E. 
Draw x a. From b l , representing the point 
of intersection between the line G P and 
the circle forming the plan of the top of 
the article, erect the perpendicular V b 
indefinitely. From a draw a perpendicular to p a, intersecting 6 1 b in the point b. Draw x b. Then x b ap will 
be the diagonal section of the article taken on the line G P of the plan. Divide the quarter circle L M, being 
the base of the cone, into any convenient number of equal parts, and from the several points carry lines per- 
pendicular to and cutting b a, as shown. From each of the points in b a erect lines perpendicular to x a and 
cutting x b prolonged, and from each of the points in x b prolonged, from x as center, describe an arc, as shown 
Establish the point 6, corresponding to M in the plan. Draw a line to the center x, as shown by i 




Fig. 322.— The Pattern of a Flaring Article, the Base of which is a Rectangle 
and the Top of which is Round, the Center of the Top being toward One 
End. 



Set the 



dividers to the space used in stepping off the plan L M, and, commencing at the point M 1 , step to the second arc, 
and from that point to the next arc, and so on, reaching the last in the point L'. Draw L' x. From M 1 as 
center, with M 1 x as radius, describe an arc, as shown by x H l , and from * as center, with radius equal to G H 
of the plan, intersect that arc in IP, as shown. Draw x W and M 1 H'. To M" IP add M 1 IT W, in all respects 
a duplicate of L 1 x W reversed. From N 1 as center, with W H' as radius, describe the arc R' V, and from H 1 



96 



Pattern Problems. 



as center, with a radius equal to H I of the plan, intersect this arc in the point I 1 . Draw H 1 I 1 and N 1 I 1 . This 
will complete one-half of the pattern, and the other several sections may be added in the same manner as 
above described. 

483. The Pattern of an Article having an Elliptical Base and a Pound Top. — Fig. 324 shows the plan 

and elevation of the article for which the pattern 
is required. The shape therein possesses some of 
the general features of a cone, but lines drawn from 
points 1, 2, 3, etc., in the base, through correspond- 
ing points 1', 2", 3 1 , etc., in the top, would reach a 
center line corresponding to the axis of a cone at 
different bights, and therefore would never meet. 
Hence, measurements must be taken upon the top and 
base direct, instead of being derived from an apex. 
Divide one-quarter part of the plan of the base into 
any convenient number of equal spaces, and divide a 
corresponding part of the plan of the top into the same 
number of spaces, by lines drawn from the points in 
the base toward the center of the circle of the top, cut- 
ting- the arc K L. Also draw the intermediate dotted 
lines connecting alternate points, as shown in the en- 
graving by 2 l 1 , 3 2 1 , 4 3 1 , etc. Construct a diagram, as 
shown by A 1 W C, Fig. 325, in which the actual dis- 
tance between corresjionding points in base and top 
shall be shown. Make C N 1 equal to the straight bight 
of the article. At right angles to it set off 1ST' A 1 , in 
length equal to the distance l 1 1 in plan. From N 1 set 
off also spaces corresponding to 2 1 2, 3 1 3, 4' 4, etc., of 
the plan, and from each of these points draw a line to 
C, as shown. Then the lines converging at C repre- 
sent the distances which would be obtained by meas- 
urements made at corresponding points uj)on the article 
itself. Construct a like diagram of the distances rep- 
resented in the dotted lines in the plan, as shown 
by C 2 N" 0, Fig. 326. Make C 2 F 1 equal to C N 
of the elevation, and from N 2 set off at right angles 
the line 1ST 2 O. Upon this line make the spaces ISP 2, 
W 3, 1ST 2 4, etc., equal to the length of the dotted lines 
l 1 2, 2 1 3, 3 1 4, etc., and from the points thus obtained 
in 1ST 2 draw lines to C\ Then these converging lines 
represent the same distances as would be obtained if 
measurements were made between corresponding 
points upon the completed article. For the pattern, 
commence by drawing any line, P X in Fig. 327, on 
which set off a distance equal to C 1 1 of the first dia- 
gram, as shown by 1 I 1 . Then, with the distance from 
1 to 2 of the plan for radius and 1 in pattern as cen- 
ter, describe an arc, which intersect by another arc 
struck from l 1 of the pattern as center and C 2 2 of the 
second diagram as radius, thus establishing the point marked 2 in the pattern. ISText, with l 1 2 1 of the plan as 
radius, and from l 1 of the pattern as center, describe an arc, which intersect by another arc drawn from 2 as 
center, and with C 1 2 as radius of the first diagram, thus locating the point 2 1 of the pattern. Continue in this 
manner, locating each of the several points shown from X to Y and from P to P of the pattern, through the 
several intersections tracing the lines of the pattern, as shown. Then X Y P P will be one-quarter of the 




Fig. 323.- 



Elevation. 



-The Pattern of an Oblong Flaring Article having a 
Round Top. 



Pattern Problems. 



97 



required 

thus com 
4Si. 



pattern. Eepeat this piece three times additional, as shown byVWTU, WXPT and YZSE 

pleting the pattern. 
The Patt t of a Flaring Article, the Top of which is Pound and the Bottom of which is OUong, 

with Semicircular Ends, the Center of the Top leing Located near One 
End,— In Fig. 328, let G K C B be the side elevation of the required 
article, of winch AEEDFPis the plan at the base and Z X I T 
the plan at the top. By inspection it will be seen that the article may 
be resolved into two frustums of cones connected by two fiat triangular 
pieces. Produce B G of the elevation indefinitely in the direction of 
H, and intersect it in the point H by a line drawn perpendicular to the 
base B C from the point L, which corresponds to the center of the top 
of the article, all as shown by L H. Then H is the apex of a cone, of 
which IT B L is a half elevation, and that portion of the article 
represented by G M L B is a frustum. In like manner produce 




12 3 4 5 6 7 



N * 3 * ■ 67 fj2 

Figs. 325 and 326 —Diagrams of Triangles. 





91 p f o 

Fig. 32S. — The Pattern of a Flaring Article, the Top of which is Round and the 
Bottom of which is Oblong, with Semicircular Ends, the Center of the Top being 
Located near One End. 

C K of the elevation indefinitely in the direction of T. Locate the 
point S in the base line, corresponding to the junction of the straight 
side and the semicircular end, as shown by B, and F of the plan, and 
draw a line from it to the point M already obtained, which produce 
until it meets C K extended in the point T. Then T is the apex of 
a scalene cone, of which T C S is the half elevation, and of which 
that part of the article represented in elevation by M K C S is a frus- 
tum. The remainder of the envelope of the article is in the shape of 
two equal triangles, one of which is shown by L M S of the elevation and by PTFof the plan. For the 
pattern proceed as follows : Divide the half plan R D F of the scalene cone into any number of equal spaces, 
and from each of the points erect a line perpendicular to the base B C of the article, and thence carry 



Fig. 327 —Pattern. 

The Pattern of an Article having an Elliptical 

Base and a Round Top. 



98 



Pattern Problems. 



the lines at right angles to the axis T S of the cone until they cut the side T C. Then from T as center strike 
arcs corresponding to these several points, all as shown. From T draw a straight line, as T F 1 , intersecting the 
first arc in the point F 1 . Set the dividers to the space used in stepping off the plan, and, commencing at the 
first arc in the point F 1 , step to the second, and from that point to the third, and so on, finally reaching the last 
in the point R'. Then a line, F 1 R 1 , traced through these several points will be the pattern of the bottom of 

the frustum of the scalene cone. From each of 
the -points in S C also draw a line toward T, as 
shown, cutting M Iv. From each of the points 
in M K erect a line perpendicular to the axis 
T S, cutting the side T C. From T as center 
draw arcs from these points in T K, as shown. 
From the points between F 1 and R 1 of the pat- 
tern draw ,!res toward T, intersecting the arcs 
just described. A line traced through their 
points of intersection, as shown by X X 1 , will 
form the pattern of the top of the frustum 
of the cone. To the two sides IS" F 1 and 
X 1 R 1 of the pattern thus far constructed add 
the'two triangles shown, in the following manner : 
From F 1 and R 1 as centers, and with F P of the 
plan as radius, describe arcs, and from X and X' 
as centers, with G B of the elevation as radius, 
describe arcs, cutting the former in the points E 1 
and W. Draw the connecting 
lines, as shown. Produce E 1 X 1 
in the direction of Y, making 
E 1 Y equal to the side of the 
cone, as shown by H B of the 
elevation. From Y as center, 
with radius equal to II B, de- 
scribe the arc E 1 X 1 , in length 
equal to the stretchout of the 
plan of the base, shown by 
E A P, and from its termina- 
tion, X 1 , draw a line to Y, as 
shown. From the same cen- 
ter, with radius equal to H G, 
describe an arc from X 1 , and 
continue it until it intersects 
X 2 X in the point W, which 

Fig. 329. — The Patterns of an Oblong Tapering Article, with One End Square and one End will complete the required 

Semicircular, having More Flare at the Ends than at the Sides. pattern. 

485. The Patterns of an Oblong Tapering Article, with One End Square and One End Semicircular, 
having More Flare at the Ends than at the Sides. — In Fig. 329, let L P (5 X M be the plan of the article at 
the top and EKHGF the plan of it at the bottom, and let A D C B be the side elevation. Inasmuch as the 
article is tapering in plan, the conical part of the pattern will include a little more than shown by a semicircle in 
plan. The lines showing the junction between the straight sides and the conical part are to be drawn perpen- 
dicular to the sides of the article. Therefore lay off in the plan Y P and Y X, drawn from the center Y of 
the curved part of the plan of the top of the article, perpendicular to the sides L P and M X respectively. 
And in like manner from Z, the center by which the curved part of the bottom of the article is struck, draw 
Z G and Z K. Draw L' P 1 and M 1 X 1 parallel to the sides L P and M X, and equal in length to them, as 
determined by the T-square placed at right angles to them and brought against their ends, and at a distance from 
them equal to the flaring bight of the side, as shown by S U of the elevation. Connect L 1 E and P 1 K, M 1 F 




Pattern 
of End. 



Pattern Problems. 



99 




and X 1 G, as shown, which will complete the patterns of the sides. For the pattern of the square end, make 
Y I equal the slant hight of the end, as shown by A B of the elevation. Through I draw M 2 I/, in length 
equal to M L, determined by the T-square, as already described in connection with the sides. Connect i/ E 
and W F. The rounded end of the article is the section of a cone, of which we must first complete the 
elevation by producing the side D C of the article in the direction of X indefinitely. From the points Gr and 
X in the plan of the bottom and top of the article respectively, drop points on to corresponding lines in the 
elevation, all as shown by G" X', through which draw a line, which produce until 
it meets D C extended in the point X. Then X is the apex of the cone, of which 
G 2 X 3 D C of the elevation is a frustum. Divide the plan X O P of the cone 
into any convenient number of equal parts, and from them let fall perpendiculars 
to the base X s D, from which carry lines at right angles to the axis X JSP, produc- 
ing them until they cut the side X D, as shown. From X as center, strike arcs cor- 
responding to the points thus obtained in X D. From the points in X 3 D carry lines 
toward the apex X, cutting G 2 C as shown, from the points in which also carry 
lines perpendicular to X X 3 , cutting X D. From X as center strike a similar set 
of arcs, as shown. From X draw any straight line, as X P 2 , producing it until it 
meets the first arc of the set representing the top of the article, as shown in the 
point P 2 . From P 2 as a starting point, with the dividers set to the same space as 
used in subdividing the plan, step to the second arc, and thence to the third, 
and so on according to the number of spaces in the plan, terminating in the point 
X 2 . A line traced through these points, as shown by P 2 X 2 , will be one side of the 
end piece. From these same points draw lines in the direction of X, crossing the 
second set of arcs. A line traced through the several points of intersection 
thus formed will be the other side of the pattern. 

4SG. The Envelope of a Eight Cone.— In Fig. 330, let ABC be the eleva- 
tion of the cone and D E F the plan of the same. Set the compasses to 
the space B A, or to the slant hight of the cone, for a radius, and from any con- 
venient point as center, as B 1 in Fig. 331, strike an 
arc indefinitely. Connect one end of the arc with 
the center, as A 1 B 1 . With the dividers step off the 
circumference of the plan D E F, as shown, and 
count the spaces until the whole or exactly one- 
half is completed. Then set off on the arc A 1 C 
the same number of steps as is contained in the 
whole plan, commencing at A 1 , which point has 
been connected with the center, as explained, and 
ending at C. Draw B 1 C. Then B' A' C 1 will be 
the envelope of the cone. 

4S7. The Envelope of a Right Cone, from 
which a Section is Cut Parallel to its Axis. — Let 
B A F in Fig. 332 be a right cone, from which a 
section is to be cut, as shown by C D in the eleva- 
tion. Let B 2 L H K be the plan of the cone. Then the line of the cut in plan is shown by D' D\ For the 
patterns proceed as follows : Divide that portion of the plan corresponding to the section to be cut off, as shown 
by D 4 B 2 D 3 , into as many spaces as are necessary to give accuracy to the pattern, and divide the remainder of 
the plan into spaces convenient for laying off the stretchout. From any convenient center, as A, with radius 
A B, describe an arc, as M X, which make equal to the stretchout of the plan B 2 L II K, dividing M N into 
the same spaces as employed in the plan. From the points in the arc corresponding to that portion of the plan 
indicated by D 4 B 2 D 3 — namely, 8 to 10 inclusive — draw lines to the center A, upon which to set off distances 
measured from the elevation. From the same points in the plan carry lines vertically, cutting the base of the 
cone, as shown from B to D, and thence continue them to the apex A, cutting C D as shown. From the 
points in C D carry lines at right angles, cutting the side of the cone, as shown in the points between C and B. 
From A as center, with radii corresponding to the points between C and B, cut the corresponding lines drawn 



Fig. 330. — Elevation and Plan. 




Fig. 331 — Pattern. 
The Envelope of a Right Cone. 



100 



Pattern, Problems. 



Pattern. 



from the same points in the stretchout to A, and through the points of intersection thus obtained trace a line, 
as shown by D 1 C D\ Then the space indicated by D 1 C D" is the shape to be cut from the envelope MAN 
of the cone to produce the shape, as shown by C D in the elevation. 

48S. The Envelope of a Frustum of a Bight Cone. — The principle involved in cutting the pattern for the 
frustum of a cone, is precisely the same as that for cutting the envelope of the cone itself. The frustum of a 

right cone is a shape which enters so extensively into articles of 
tinware, that we have thought it well to illustrate the cutting of a 
pattern for it by an engraving of somewhat different character 
from those employed in other and similar cases. In Fig. 333 is 
shown by plan and elevation an ordinary flaring pan, which is an 
illustration of the articles employing the shape of a frustum of a 
cone. For the pattern proceed as follows : Through the eleva- 
tion draw a center line, K B, indefinitely. Extend one of the 
sides of the pan, as, for example, D O, until it meets the center 
line in the point B. Still greater accuracy will be insured by 
extending the opposite side of the pan also, as shown — the three 
lines meeting in the point B — which determines the 
apex of the cone to a certainty. Then B O and 
B D respectively are the radii of the arcs which 
contain the pattern. From B or any other conveni- 
ent point as center, with B as radius, strike the 
arc P Q indefinitely, and likewise from the same 
center, with B D as radius, strike the arc E F indefi- 
nitely. From the center B draw a line across these 
arcs near one end, as P E, which will be an end of 
the pattern. By inspection and measurement of the 
plan, determine in how many jfieces the plan is to 
be constructed, and divide the cii*cumference of the 
plan into a corresponding number of equal parts, 
in this case three, as shown by K, M and L. With 
the dividers or spacers step off the length of one of 
these parts, as shown from M to L, and set off a 
corresponding distance on the arc E F, as shown. 
Through the last division draw a line across the 
arcs toward the center B, as shown by F Q B. Then 
PQFE will be the pattern of one of the sections 
of the pan, as shown in the plan. In the engraving 
the plan has been placed below and in line with the 
elevation, in order to better show the correspond- 
ence of parts in the two views. In practice this is 
not necessary ; but, since in most cases it can be as 
well placed there as anywhere else, it is advisable to 
do so on account of the greater accuracy insured by 
drawing lines through corresponding points with 
the T-square, as illustrated by D IT, etc. Neither 
In this demonstration, as well as in many others 

the pattern, but have 




Elevation 



Fig. 332. — The Envelope of a Right Cone, from which a Section, is 
Cut Parallel to its Axis. 



is it necessary to draw more than one-half of the elevation. 

in this book, we have not limited ourselves to the smallest number of lines for describing 

put in enough others to show the reason for every step taken. 

489. The Envelope of the Frustum of a Right Cone, the Upper Plane of which is Oblique to its Axis. — In 
Fig. 334, let C B D E be the elevation of the required shape. Produce the sides C B and E D until they inter- 
sect at A. Then A will be the apex of the cone of which C B D E is a frustum. Draw the axis A G, which 
produce below the figure, and from a center lying in it draw a half plan of the article, as shown by F G H. 
Divide this plan into any number of equal parts, and from the points carry lines parallel to the axis until they 



Pattern Problems. 



101 



cut the base Hue, and from there extend them in the direction of the apex until they cut the upper plane B D. 

Place the T-square at right angles to the axis, and, bringing it against the several points in the line B D, cut the 

side A E, as shown. From A as center, 
with A E as radius, describe the arc 
C E 1 , on which lay off a stretchout of 
either a half or the whole of the plan, 
as may be desired, in this case a half, as 
shown. From the extremities of this 
stretchout, C and E 1 , draw lines to the 
center, as C A and EA. Through the 
several points in the stretchout draw sim- 
ilar lines to the center A, as shown. 
"With the point of the compasses set at 
A, bring the pencil to the point D in the 
side A E, and with that radius describe 
an arc, which produce until it cuts the 
corresponding line in the stretchout, as 
shown at D 1 . In like manner, bringing 




Planl 



Fig. 333-" 



-The Envelope of a Fi~u,stum of a 
Right Cone. 




the pencil against the several points 
between D and E in the elevation, 
describe arcs cutting the correspond- 
ine measuring lines of the stretch- 
out. Then a line traced through 
these intersections will form the 
upper line of the pattern, the pat- 
tern of the entire half being con- 
tained in C B 1 D 1 E'. 

490. The Envelope of a Scalene 
Cone. — The difference between a 
scalene cone and a right cone con- 
sists of the base line. In one it is 
drawn oblicpie to the axis, while in 
the other it is at right angles to the 
axis. In Fig. 335, let G D II be Fig. 
the elevation of a scalene cone, the 
pattern of which is to be cut. At right angles to the axis D O, and through the point G, draw the line E i. 
Extend the axis, as shown by D B, and upon it draw a plan of the cone as it would appear when cut upon the 



334.— The Envelope of the Frustum of a Right Cone, Vie Upper Plane of which 
is Oblique to its Axis. 



102 



Pattern Problems. 



line E F, as shown by A B C. Divide the plan into any convenient number of equal parts, and from the points 
thus obtained drop lines on to E F. From the apex D, through the points in E F, draw lines to the base G H. 
From D as center, with D G as radius, describe an arc indefinitely, on which lay off a stretchout taken from 
the plan ABC, all as shown by I M K. From the center D, by which the arc was struck, through the points 
in the stretchout, draw radial lines indefinitely, as shown. Place the blade of the "["-square parallel to the line E F, 
and, bringing it against the several points in the base line, cut the side D II, as shown from F to H. With one 
point of the compasses in D, bring the other successively to the points 1, 2, 3, 4, etc., in F H, and describe arcs, 
which produce until they cut the corresponding lines drawn through the stretchout, as indicated by the dotted 
lines. Then a line, ILK, traced through these points of intersection, as shown, will complete the required 
pattern. 

491. The Envelope of a Frustum of a Scalene Cone, or the Envelope of the Section of a Right Cone, con- 
tained between Planes Oblique to its Axis. — In Fig. 336, let F L M K represent the section of the cone the 

pattern for which is required. Produce the 
sides F L and K M until they meet in the 
point N, which is the apex of the cone of 
which F L M K is a frustum. Through E" 
draw the axis of the cone, which produce in 
the direction of D indefinitely. .From K draw 
K II at right angles to the axis. At conveni- 
ent distance from the cone, either above or 
below it, construct a plan or profile as it 
would appear when cut on the line K H, let- 
ting the center of the profile fall upon the 
axis produced, all as shown by A D C B. Di- 
vide the profile into any number of equal 
parts, and from the points thus obtained draw 
lines parallel to the axis, cutting K EL From 
the apex N, through the points in ~K H, draw 
lines cutting the top L M and the base F K. 
Place the blade of the T-square at right angles 
to the axis of the cone, and, bringing it suc- 
cessively against the points in L M and F K, 
cut the side N F, as shown above L, and from 
H to F. From 1ST as center, with radius N H, 
strike the arc T S indefinitely, upon which lay 
off a stretchout from the plan, as shown, and 
through the points from the center 1ST draw 
lines indefinitely, as shown. With the point 
of the compasses still at N, and the pencil 
Fig. 335-— The Envelope of a Scalene Cone brought successively against the points in the 

side from H to F, describe arcs, which produce until they cut corresponding lines drawn through the stretchout. 
Then a line traced through these points of intersection, as shown by T U S, will form the lower line of pat- 
tern. In like manner draw arcs by radii corresponding to the points in the side at L, which produce also until 
they intersect corresponding lines drawn through the stretchout. A line traced through these points, asEPO, 
will be the upper line of the pattern sought. 

492. The Envelope of the Frustum of a Cone, the Base of which is an Elliptical Figure. — This shape is 
very frequently used in pans and plates, and therefore we have employed the representation of an ordinary oval 
or elliptical pan in our engraving by way of illustration. (See Fig. 337.) Draw an elevation of either a side 
or end of the article, and corresponding to and in line with it lay off the plan, as shown, employing for this 
purpose any rule for constructing the ellipse which employs centers. In Fig. 337, let that part of the plan 
lying between H and L be an arc whose center is at TJ, and let those portions between V and H and L and W 
be arcs whose centers are respectively K and S. In drawing the plan, let it be composed of two lines, one of 
which shall represent the plan of the vessel at the top and the other the plan of the vessel at the bottom. 




Pattern Problems. 



103 



ACDB represents an elevation of the vessel, and is so connected with the plan as to show the relation- 
ship of corresponding points. After having drawn the plan, the next step is to construct the diagram shown 
in Fig. 33S. Draw the horizontal line II U indefinitely, and at right angles to it draw H A, indefinitely also. 
Make II IT, Fig. 338, equal to H U of the plan, 
Fig. 337. Make H C of Fig. 33S equal to the 
vertical hight of the vessel, as shown in the ele- 
vation by D X. Draw the line C G parallel to 
Ii U, making C G in length equal to U X 1 of the 

the 



plan, Fig. 337. 




Through 



Eig 337— Elevation and Plan. 
A 




Fig 



R U 

33S.— Diagram of Radii. 



the points 
IT and G thus estab- 
lished draw the line 
U G, which continue 
until it meets H A in 
the point A. Then 
A U will be the radius 
by which to describe 
that portion of the 
pattern which is in- 
cluded between the 
points II and L of the 
plan. With A U as 
radius, and from any 
convenient point as 
center — as, for exam- 
ple, A, Fig. 339— 
draw the arc H L, 
which in length make 
equal to H L of the 
plan, Fig. 337, as 
shown by the points 
1, 2, 3, etc. From the 
same center, and with 




Elevation. 



Fig. 336. — The Envelope of a Frustum of a Scalene Cone. 

the radius A G of Fig. 33S, describe the parallel arc X O. From the points 
H and L of the arc first drawn, draw lines to A, thus intercepting the arc 
X O and determining its length. In the diagram, Fig. 338, set off from H, 
on the line II U, the distance II B. making it equal to B II of the plan, Fig. 
337. Also, upon the line C G, from the point C, set off C I, equal to E X 
of the plan, Fig. 337. Then, through the points B and I thus established, 
draw the line E B, which produce until it intersects A II. Then E B will 
be the radius for those portions of the pattern lying between Y and H and 
L and W of the plan, Fig. 337. From the point II, on the line H A, Fig. 
339, set off the distance H B, equal to E B of Fig. 338. Then, with B as 
center, describe the arc E II, and from corresponding center C, at the oppo- 
site end on pattern, describe the arc L K. From the same centers, with B I 
as radius, describe the arcs X M and P, all as shown. Make H E and 
L K in length equal to H E and L K of the plan, Fig. 337. From E and 
K, respectively, draw lines to the centers B and C, intercepting the arcs 
X M and Pin the points M and P. Then E K P M will be one half of 
the complete patterns of the vessel. 

493. The Pattern of a Flaring Article which Corresponds to the Frus- 
tum of a Cone whose Base is a True Ellipse. — In Fig. 340, let G II F E be the elevation of one side of the 
article, L M U E the elevation of an end, E 1 E 1 F 1 U 1 the plan of the article at the base, and T V S P the plan 




Eig 339 — Pattern. 
The Envelope of the Frustum of a Cone, 
the Base of which is an Elliptical 
Figure. 



104: 



Pattern Problems. 



at the top. Produce E G and F H of the side elevation until they meet in the point I. At any convenient 
place draw the straight line D A of Fig. 341, in length equal to I E\ Make D B equal to I G 1 . From A and 
B draw perpendiculars to D A indefinitely, as shown by A O and B K Divide one-quarter of the plan E 1 E 1 
into any convenient number of equal parts, as indicated by the small figures. From the points thus determined 

draw lines to the center C, and also carry lines perpen- 
dicular to the base of the article E F, as shown, from 
which line continue them toward the apex I, cutting the 
top G H, as shown. Take the distances C 5, C 4, C 3, 
etc., of the plan and set off corresponding distances from 
A on A 0, as sho^vn by A 5, A 4, A 3, etc. From these 
points in A O draw lines to D, cutting B 1ST. From D 
as center, describe arcs corresponding to the several points 
in A O, as shown. From any convenient point in the 
first arc draw a straight line to D, as shown by "W D. This 
will form one side of the pattern. From W as a starting 
point, lay off the stretchout of the plan E 1 , B, 1 , F 1 , etc., 
irsing the same length of spaces as employed in dividing it, 
stepping from one arc to the next each time, as shown. A 
line traced through these points will be the outline of the 
plan, one-half of the entire enveloj)e being shown in the 
pattern from "W to Z. From these points also draw lines 
to the center D, and from D intersect them by arcs drawn in the 
same manner as before described, corresponding to the several points 
in B X. A line traced through the points of intersection between 
these arcs and the radial lines from D will form the upper line of 
the pattern, as shown. Then ¥XTZ will constitute the pattern 
of one-half of the envelope, to which add a duplicate of itself for 
the complete pattern. 

494. Patterns of a Tapering Article with Equal Flare through- 
out, which Corresponds to the Frustum of a Cone the Base of which 
is Elliptical (Struck from Centers), the Upper Plane of the 
Frustum being Oblique to the Axis. — In Fig. 342, let H F G A be 
the shape of the article as seen in side elevation. The plan is shown 
by I L IS" 0. In order to indicate the principle involved in the 
development of this shape, we have introduced lines which show 
the construction of the figure. It may be remarked at the outset 
that a conical figure having an elliptical base, or, in other words, 
whose base corresponds to a figure struck from centers, exhibits 
throughout its extent the peculiar properties of its base. In other 
words, a conical solid, the base of which is an elliptical figure struck 
from various centers, resolves itself into sections of cones, the seve- 
ral bases of which correspond to the circles, arcs of which compose 
the elliptical base. Thus, by inspection of the engraving, it will be 
seen that the shape H F G A is made up of sections of cones cor- 
responding to the arcs of which the plan I L 1ST O is composed. Those parts of the figure shown in plan by 
KUTM and KUTP may be considered as segments cut from a right cone, the radius of the base of which is 
either O K or L B, and the apex E of which is to be ascertained by producing the lines K 1 D and M 1 B, through 
the points D and B, until they meet in E. By further examination it will be seen that the points K 1 and 
M l are established by vertical lines carried to the base from the points K and M of the arc struck from the cen- 
ter 0, and the points D and B, representing the points of intersection above referred to, fall somewhere in ver- 
tical lines corresponding to U T of the plan. These points of intersection between the lines drawn from K' 
and M 1 , and U and T, are determined by producing the sides H F and A G of the given figure until they meet 
the vertical lines drawn from U and T. The parts shown in the plan by K U R and M T P may be con- 




Fig. 341.— LHagram of Triangles and Patterns. 

Tlie Pattern of a Flaring Article which corre- 
sponds to the Frustum of a Cone ivhose Base 
is a True Ellipse. 



Pattern Problems. jqk 

sidered as segments cut from a right cone, the radius of the base of which is either IT I or T N The apex of 
this cone is to be found by means of an end elevation, in which are drawn hues corresponding to the points R 

and K of the plan, to B', and which in hight correspond to the 
points B and D of the side elevation already determined, all 
as shown in E a B' and K s B 1 . The conditions of this figure, 
as stated, are that it shall have equal flare throughout; there- 
fore, the pitch of the sides 0° C and 1/ C in the end elevation 
must be equal to those of the side elevation, determined by 
the established flare of the article. By this 
means the point C in the end elevation is 
determined. From C the point C in the 
side elevation is derived, as indicated by the 




dotted line connecting the two. By further ex- 
amination of the engraving it will be seen that 
the part shown by KUTM in plan is shown in 
elevation by K 1 D B M 1 , and that the part shown by 
R IT K in plan is shown in elevation also by R 2 B 1 
K 3 , the apexes of the small cones forming the ends 
of the figures falling at D and B, while the apex 
of the large cone, from which the two middle sec- 
tions are cut, is at a higher point, shown at E. 
„. ' ' an ' This condition of things gives rise to the shape D 

Fig. 342.— Patterns of a Tapering Article with Equal Flare through- „ ,-, , . , , ° . ... i 

out, which Corresponds to the Frustum of a Cone the Base of C L > aS 8hown m Slde elevation, which corresponds 
which is Elliptical (Struck from Centers), the Upjier Plane of to U T of the plan, and which ill end elevation 

the Frustum being Oblique to the Axis. is shown by C B', being a parabolical curve. It is 

formed by the sections of the larger cone, shown in end elevation by O 2 V C and 1/ V C 1 , meeting on the line 
C 1 V. In connection with the side elevation, by means of the lines L" E 1 0", is shown a vertical section of one- 
half of the larger cone from which the segments are cut. Thus it will be seen that the base O 2 U corresponds 
to O L of the plan, and the apex E 1 corresponds to the apex E of the side elevation. By comparing section 
If V C 1 with this larger figure, of which it is a part, the nature and construction of the shape will be more 
clearly seen. K 3 V B 1 in the end elevation represents a corresponding section of the smaller cone, the side 
K? B 1 of which, being produced, meets the side of the larger cone in the point E 1 . This indicates a correspond- 
ence of parts which admits of the figure being constructed in the way we have specified. Having thus described 
the nature of the figure, the manner of drawing the two elevations, both of which are necessary in developing 
the patterns, becomes evident without further explanation. For the patterns we proceed as follows: 



106 



Pattern Probl 



ems. 



Divide one-half of the plan into any convenient number of equal parts, as shown by the small figures, 
and from the points thus established carry lines vertically, cutting the b ise line II A, and thence carry 
them toward the apexes of the various cones from the bases of which they are derived. That is, from the 
arc K M draw lines toward the apex E, and from the points derived from the arc I K carry lines toward 
the apex D, and in like manner from the points derived from the arc M N" carry lines in the direction of 
the apex B, all of which produce until they cut the top line F CI of the article. From the points in F G 
thus established carry lines to the right, cutting the slant lines of the cones to which they correspond. Thus, 
from the points occurring between F and f, draw lines cutting B A, being the slant of the small cone, as shown 
by the points immediately below "W. In like manner, from the points between g and G, carry lines cutting 
the same line, as shown by G. The slant line of the large cone is shown only in end elevation, and therefore 
the lines corresponding to the points between f and g must be carried across until they meet the line B 1 L*. 
Commence the pattern by taking any convenient point, as E', for center, and E' L" as radius, and strike the arc L 2 S 
indefinitely. Upon this arc, commencing at any convenient point, as K 4 , set off that part of the stretchout of 
the plan corresponding to the base of the larger cone, as shown by the points 5 to 13 in the plan, and as indi- 
cated by corresponding points from K 4 to M 2 in the arc. From the points thus established draw lines indefi- 
nitely in the direction of the center E 1 , as shown. From E' as center, with radii corresponding to the points 5 to 

13 inclusive, established in the line B 1 L 2 already described, cut correspond- 
ing radial lines just drawn, and through the points of intersection thus es- 
tablished draw a line, all as shown by f g'. Next take A B of the side 
elevation as radius, and setting one foot of the compasses in the point K 4 of 
the arc, establish the point D 1 in the line K 1 E 1 , and in like manner, from 
M 2 , with the same radius, establish the point B 2 in the line M 2 E 1 , which 
will be the centers from which to describe those parts of the patterns derived 
from the smallei^one. From D 1 and B 1 as centers, with radius B A, strike 
arcs from lv 4 and M 2 respectively, as shown by K 4 I 2 and M 2 W, upon which 
set off those parts of the stretchout corresponding to the smaller cones, as 
shown by the arcs K I and M N of the plan. From the points thus estab- 
lished, being 5 to 1 and 13 to 17 inclusive, draw radial lines to the centers 
D' and B 2 , as shown. For that part of the pattern shown from F 1 to f, 
set the dividers to radii, measuring from B, corresponding to the several 
points immediately below W of the side elevation, and from D' as center 
cut the corresponding radial lines drawn from the arc. In like manner, for 
that part of the pattern shown from G' to g\ set the dividers to radii meas- 
ured from B, corresponding to the points in the line B A at G, with which, 
from B 2 as center, strike arcs cutting the corresponding measuring lines, 
as shown. Then F 1 G 1 1ST I 2 will be one-half of the pattern sought — in 
other words, corresponding to I K L M N of the plan. The whole pat- 
tern may be completed by adding to it a duplicate of itself. 
495. The Pattern of an Irregular Flaring Article, botJi Top ana 'Bottom .of 'which are Pound, the Top 
heing Smaller than the Bottom, and the two being Tangent at One Point in Plan.— In Fig. 343, let B D E C 
be the side elevation of the article, one-half of the plan of the bottom being shown in F II G, and one-half 
of the plan of the top by F K I. For the pattern proceed as follows : Produce the side E D indefinitely in the 
direction of A. Produce the side C B until it meets the other in the point A. Having the plan drawn 
directly in line with the elevation, so that like points in each correspond, all as shown in the engraving, 
divide the plan of the base and the plan of the top into the same number of equal spaces, as shown by 
a 1 , b\ c 1 , d\ etc., and a, b, c, d, etc., respectively. This may be done by dividing the base and cutting the circle 
of the top bylines drawn from these points to the point F. From F as center, with F a 1 , F I 1 as radii, describe^ 
arcs, as shown, cutting F G. From F G continue them at right angles until they cut the base C E, whence 
carry them toward the apex A, cutting the top B D. From any convenient point, as A' in Fig. 344, as 
center, with radius A E of Fig. 343, describe an arc, as shown by G 3 G\ In like manner, with radii A i\ 
A li, A g\ etc., describe arcs indicated by i 3 i% h* h\ g' g\ etc., in the pattern. From the same center A 1 , with 
corresponding radii taken from A of the elevation, to the intersections made by the radial lines with the top 




Fig. 343.— Elevation and Plan. 
An Irregular Flaring Article, both Top 
and Bottom of which are Round, the 
Top being Smaller than the Bottom, 
and Tangent at One Point in Plan. 



Pattern, Problems. 



107 



B D, describe arcs, as shown in the pattern by i~ i~, g 2 g°~, h 2 h", etc. Draw any straight line from A' to the first 
arc corresponding to the points in the base, as shown by A 1 C, which will represent one side of the required 
pattern. Set the dividers .to the space used in stepping off the plan of the base, and, starting witli C, lay off 
the stretchout, stepping from arc to arc, as shown. Trace a line, C E 1 C 1 through these points, which will be 
the bottom of the required pattern. From these same points draw lines to the center A', cutting the set of 
smaller arcs. Trace a line, B 1 D' B 1 , through the intersections of these lines with arcs of corresponding numbers, 
which will be the top line of the pattern. From the last points in the line C 1 E 1 C draw a line toward the 
center A 1 , as shown, reaching B', which will complete the pattern. 

496. The Pattern for an Irregular Flaring Article which in Elliptical at the Base, Pound at the Top, 
the Top being so Situated with Pespect to the Base as to be Tangent to One End of it when Viewed in 
Plan. — In Fig. 345, let D 6 F E be the side elevation of the article and K N M onedialf of the plan of the 
base. The half plan of the top is shown by K W L, the base and top being tangent in plan at the point K. 
The pattern for this shape is to be obtained by cutting the surface up into triangles so small that there is 
no apparent curve between points. To do this, proceed as follows : Divide the plans of the top and base 
into the same number of equal parts, 
as shown by 1, 2, 3, etc., in the base 
and l 1 , 2 1 , 3 1 , etc., in the top, and 
connect similar points in the two 
by lines, as shown by 6 6', 5 5', etc. 
Also connect each point in the plan 
of the top with the next lower 
number in the plan of the base, as 
shown by the diagonal dotted lines 
in the engraving, as G f , 5 C, etc. 
At any convenient point draw A IT, 
in length equal to D E of the eleva- 
tion, and lay off U T at right angles 
to it. Let A represent all points 
in the circle which is the plan of 
the top of the article. Lay off from . 
U the distance from each of the 
several points in the circle to the 
corresponding point in the ellipse. 
Thus make II 7 equal to 7' 7 of the 
plan, U 6 equal to 6' G, etc. Draw 
the radial lines A 2 , A 3 , A 4 , etc. In 
like manner construct a corresponding section, as shown by C B V, using for the spaces in B Y the length 
of the diagonal or dotted lines between the circle and the ellipse in the plan. Draw C 2 , C 3 , C 4 , etc. By means 
of these two sets of lines, converging at A and C respectively, we have the actual dimensions of the triangles 
into which we have imagined the surface of the article to be divided, and which in plan are shown by 7 V G, 
7' 6 6', 6 6 1 5, etc. These are to be used in describing the pattern as follows: At any convenient place draw 
the straight line P B in Fig. 346, in length equal to G F of the elevation, or, what is the same, equal to A 7 of 
the first diagram. As we have shown but half of the plan, the pattern will also appear as one-half of the whole 
shape, and therefore P B will form its central line. From P as center, with radius C 6 of the second diagram, 
describe an arc, which intersect by a second arc struck from B as center, with radius 7 6 of plan, thus establish- 
ing the point 6 of the pattern. Then with radius A 6 of the first diagram, from 6 of the pattern as center, 
describe an arc, which cut with another arc struck from 7 1 of the pattern as center, and 7' 6' of the plan as 
radius, thus locating the point 6 l of the pattern. Continue this process, locating in turn 5 5 1 , 4 4', etc., until 
points corresponding to all the points laid off in the plan are established. Draw lines through these points. 
Then P B S will be one-half of the required pattern. 

497. Pattern for a Scale Scoop. — In Fig. 347, let AB CD represent the side elevation of a scale 
scoop, being a style in quite general use, and E F II Gr a section of the same as it would appear cut 




Fig. 344.- Pattern. 

An Irregular Flaring Article, both Top and Bottom of which are Round, the Top 
being Smaller than the. Bottom, and the two being Tangent at One Point in 
Plan. 



108 



Pattern Problems. 



Fig. 345.— Elevation. Plan and Diagrams of Triangles. 



upon the line B D, or, what is the same, so far as concerns the development of the patterns, an end 
elevation of the scoop. The following rule also applies to other forms. The curved line ABC, representing 

the top of the article, may be 
drawn at will, being, in this 
case, a free-hand curve. For the 
patterns proceed as follows: 
From the center K, by 
which the profile of the 
section or end elevation 
is drawn, draw a horizon- 
tal line, which 
produce until it 
meets the center 
line of the scoop 
in the point 0. 
Produce the line 
of the side D C until it meets the line just drawn 
in the point X. Then X is the apex and X the 
axis of a cone, a section of the envelope of which 
each half of the scoop may be supposed to be. 
Divide one-half of the profile, as shown in end ele- 
vation by E G, into any convenient number of 
spaces, and from the points thus obtained carry 
lines horizontally, cutting the line B D, as shown, 
and thence carry lines to the points X, cutting 
the top B C, as shown. With X D as radius, and 
from X as center, describe an arc, as shown by L E", 
upon which lay off the stretchout of the scoop, as 
shown in end elevation. From the points in LN 
thus obtained, draw lines to the center X, as shown. 
From the points in B C, formed by the lines drawn 
from B D to the point X, drop lines cutting the 





Fig. 346.— Pattern. 



The Pattern for an Irregular Flaring A) tide 
which is Elliptical at the Base, Round at the 
Top, the Top being so Situated with Respect 
to the Base as to be Tangent to One End of 
it when Viewed in Plan. 




side D C, as shown. With X as cen- 
ter, and radii corresponding to each of 
the several points between D and 0, 
describe arcs, which produce until they 
cut radial lines drawn from the arc Fi 9- 347-— Pattern for a Scale Scoop. 

L N to the center X of corresponding numbers. Then a line traced through the points thus obtained, as 
shown by L M IST, will be the profile of the pattern of one-half of the required article. 

498. An Irregular Section through an Elliptical Cone. — In Fig. 348 is shown an irregular section cut 
from a cone, the base of which is elliptical. Forms somewhat similar to this are in use for various purposes. 



Pattern Problems. 



109 



Side 
Elevation. / 



Without naming a list of the articles in which the principles here explained are used, we will present a single 
demonstration. Application of the same principles may be made in constructing similar shapes to the 
one here illustrated, for whatever use required. In the engraving BE'CD is the plan of the cone from 
which the irregular section, shown by r o x s, is cut. An end elevation of the cone and also of the article 
required is shown in E l A 2 D 1 . A E of the side elevation is the center of the shape. By inspection of 
the several views it will be seen that three patterns are required : the top, or cover, the bottom and the rim. 
The stretchout of the bottom is obtained 
by stepping off the length of the line * x 
in the side elevation and laying the same 
down on B 2 C", as shown in the pattern. 
Through the points in the line B 2 C 2 thus 
obtained draw measuring lines in the 
usual manner. Since B E ! C D of the 
plan represents a straight section through 
the cone., on the line F G, and as the 
shape of the article we are seeking is a 
curved or warped surface cutting through 
the cone above the base, we cannot use 
the plan B E 2 C D in laying off the width 
of the bottom, but must obtain a line in 
it corresponding to the continuous point 
of contact made between the edge of the 
bottom face and the rim. To do this we 
proceed as follows : Divide the quarter 
of the plan, as shown by E 2 C, into any 
convenient number of equal parts, as 
shown by the small letters, a, b, c, etc. 
From these points carry iines vertically 
to the base line E G of the cone, and 
thence continue them toward the apex 
A, crossing the rim, as shown in the 
side elevation. From the center I of the 
plan draw lines to those same points, a, 
b, o, etc. From the points formed in the 
line s x of the article drop points back 
on to the plan, cutting the radial lines of 
corresponding numbers. Through the 
points of intersection thus obtained trace 
a line, as shown by the second line in the plan, which will represent the line of contact between the envelope of 
the cone and the bottom of the rim, as seen in plan. Upon the several measuring lines drawn through the 
stretchout B 2 C 2 , set off on either side the length upon lines of corresponding numbers, drawn from the second 
line in the plan to the center line I C. Through the points thus obtained trace a line, as shown by B 2 E° C 2 D 3 , 
which will be the pattern of the bottom of the article. The pattern of the top is obtained in the same way. 
Drop the points back into the plan from the line B. 0, thus obtaining the inner line in the plan, which repre- 
sents a continuous point of contact between the envelope of the cone and the top of the article. Lay off a 
stretchout of r o, as shown by B' C, through which draw measuring lines in the usual manner, and upon these 
lines set off distances, measured on corresponding lines in the plan, from the center line I C to the inner line 
just obtained, by dropping points from R 0, as shown. Then B 1 E 5 C D 2 will be the pattern of the upper 
piece. For the pattern of the rim we proceed as follows : Produce the base line F G of the cone indefinitely, 
as shown from E 3 , upon which erect a perpendicular, E 3 A 3 , at any convenient place, in length equal to 
A E of the side elevation. Drop all the points in the line /' o and s x on to A 3 E 3 , by lines carried at right 
angles to the axis A E, and from these points in A 3 E 3 produce lines indefinitely, as shown. Upon the base line 
F G prolonged, measuring from E 3 , set off the length of the radial lines in the plan, measuring from I to the 




D 3 
Pattern of Top. Pattern of Bottom. 

Fi(j. 348. — An Irregular Section through an Elliptical Cone. 



110 



Pattern Problems. 




outer line of the plan, as shown, thus making E 3 A 1 equal to I a, and E a b 1 equal to I 5, etc. From the points 
a', b\ c\ d\ etc., thus obtained, draw lines to A 3 , cutting both sides of horizontal lines last described. From A 3 
as center draw arcs corresponding to the points a 1 , V, c\ d\ etc., as shown. Set the dividers to the space used 
in stepping off the plan E 3 C, and lay off its stretchout, stepping from any convenient point in the arc corre- 
sponding to «' to the arc corresponding to b\ and from this to the arc corresponding to c\ From the points 

thus obtained, indicated by « 2 , b 2 , e 3 , d*, etc., draw radial lines to A 3 
from the various points of intersection between the horizontal lines 
drawn from A 3 E 3 . "With the radial lines drawn from the points 
a', b', e', d\ etc., describe arcs, which produce until they cut the 
lines of corresponding number drawn from the points a', V, e 3 , etc. 
Through the points of intersection thus obtained trace lines, as 
shown by r 1 o l and x 1 «'. Then r l cr s l x 1 is the pattern of one- 
quarter of the rim. 

499. Patterns for a Hip Bath.— In Fig. 349, let H A L N O 
be the elevation of the bath, of which D 1 G E' B 1 is a plan on the 
line D E. Let the section A B W B 8 , Fig. 350, represent the flare 
which the bath is required to have through its sides on a line indi- 
cated by A B in elevation. By inspection of the elevation it will 
be seen that three patterns are required, which, for the sake of con- 
venience, we have numbered in the various representations 1, 2 and 
3. In the following demonstration the various parts are treated as 
sections of cones, that of No. 1 being an irregular frustum of a right 
cone, that of No. 2 the frustum of a scalene cone, while the foot, or 
No. 3, is a section of a cone whose base is oval or egg-shaped. For 
the patterns, commencing with No. 1, we proceed as follows: Pro- 
duce the line II D indefinitely, and also A B, likewise indefinitely, 
until they meet in the point F. Then F is the apex of the cone of 
ng. 349-— Elevation and plan. which No. 1 is a section. From any convenient point, as F 1 , Fig. 

Hip Bath. Z5\, for center, with F D as radius, describe an arc, as shown by 

B 4 G 1 . Divide that portion of the plan corresponding to piece No. 1, as shown by D 1 G, into any con- 
venient number of equal parts, as indicated by the small figures 1, 2, 3, etc. From the points thus obtained 
carry vertical lines cutting the base D B, as shown. From the apex F of the cone, through 
the points D B thus obtained, carry lines cutting the upper boundary II A of the piece. 
From the points in II A carry lines at right angles to the axis A F, cutting the side D H, 
as shown by the small figures. From F 1 in Fig. 351, the center by which the arc B 4 G 1 was 
described, draw a straight line indefinitely, cutting B 4 G 1 near the center, as is shown by 
F 1 IF. From D", the point at which this straight line crosses the arc, measuring both ways, 
set off a stretchout of D' G of the plan, all as indicated by the small figures. Through the 
points thus obtained in B 4 G 1 draw radial lines indefinitely. From the center F 1 upon the m P Bath - 

straight line F' IF, which it will be seen corresponds to F II of the elevation, set off points corresponding to 
the points in F H, all as indicated by the small figures, 1, 2, 3, etc. From F as center, with radii corres- 
ponding to these points, strike arcs, which produce both to the right and the left until they cut radial lines of 
corresponding numbers. Then a line traced through these points, as shown by A" II 2 A 5 , will be the 
boundary of the pattern upon its upper side, and the whole pattei-n will be contained by B 4 G 1 A 5 LP A 4 . Since 
the small diagram A 6 IP B 6 , Fig. 350, represents a section of the article upon the line A B in elevation, set 
off the angle A M B in elevation equal to W A 6 B" of the diagram, and through the points M B draw a line 
indefinitely. Produce the line L E until it meets the line drawn through M B in the point X. Then X is the 
apex and X K is the axis of the scalene cone of which M L E B of the elevation is a section. Divide that por- 
tion of the plan corresponding to this piece, No. 2, into any convenient number of equal parts, as indicated by 
the small figures, and from the points thus obtained carry lines vertically, cutting the base E B, as shown. 
From the apex X, through the points thus obtained, carry lines cutting the top of the piece L M, as shown. 
From the points in E B, and also from the corresponding points in L M, draw lines at right angles to the axis 
X K, cutting it as shown in the two sets of points marked with the small figures 1, 2, 3, 4, etc. Lay off 




Fig. 350. —Flare at 
Sides. 



Pattern Problems. 



Ill 




Fig. 3ST.— Pattern of Back (No. i). 
Hip Bath. 



X 1 K 1 , Fig. 352, at any convenient point, equal to X K of the elevation, in which set off the points corresponding 
to the points just obtained in X K, all as indicated by corresponding figures. From each set of points in X 1 K' 

erect lines indefinitely, perpendicular to X 1 K 1 , all as shown. In the 
lines drawn from the points, commencing at B 1 , set off lengths cor- 
responding to lengths measured from C of the plan, on radial lines 
drawn to the points stepped off in G E', and through the points thus 
obtained draw radial lines from F 1 , as shown, producing them until 
they cut corresponding lines drawn from the points commencing at 
M\ From X 1 as center, with radii corresponding to the intersections 
between the radial lines and the perpendiculars drawn from the 
points at B 1 and M', describe arcs indefinitely, as shown. From X 1 
draw any straight line, as X 1 M 3 , as shown, crossing the arcs just 
described, which will form a basis of measurement for one side 
of the pattern. From the point B 3 , where the line X 1 M' crosses 
the first arc corresponding to the set of points, commencing at B 3 , 
step off the stretchout of the plan G E 1 , using the same spaces as 
first employed, stejDping from arc to arc, as shown. Then a line 
traced through the points thus obtained, as shown by B 3 E 2 B 2 will 
be the edge of the pattern corresponding to B E of the elevation. 
Through the points in B 3 E 2 B 2 , from X 1 , draw radial lines, which pro 
duce until they cut arcs of corresponding numbers drawn from the points in the lines at M 4 . Then a line 
traced through these points, as shown by M 3 L 2 M s , will be so much of the line of pattern corresponding to the 
top of 'the article in elevation as shown from M to L. 
From B a and B 3 respectively as centers, with radius ecpial 
to A 5 B c of the small section to the left of the elevation, 
describe an arc, and from M 2 and M 3 as centers, with radius 
equal to M 1 A" of the small section, describe arcs intersect- 
ing those first drawn in the points A and A. Continue the 
line of the outside of the pattern from W and M 3 , re- 
spectively, to A and A 3 . Then A L 2 A 3 B 3 B 2 will be the 
pattern of No. 2. For pattern of Xo. 3, first construct 
a section of radii, as shown in Fig. 353 of the engrav- 
ings. The plan corresponding 
to the line D E of Xo. 3 in ele- 
vation, as shown by E 1 ' G E 1 
B 1 , is struck from several cen- 
ters. From C a semicircle, B 1 
D" G, is struck. Arcs G R and 
B 1 R 1 are struck from centers P 
and P 1 , and the arc P P 1 is 
struck from the center S. There- 
fore, to obtain the radii by which 

the pattern may be described, we must construct sections of the several cones 
which the arcs composing the plan may be supposed to belong. Draw any 
line, as O 1 T, indefinitely, at right angles to which set off O" X 1 indefinitely. From 
O 1 , measuring on the line O 1 T, set off O l G equal to the vertical higlit of piece 
Xo. 3 measured in elevation, and from C 1 draw G E' perpendicular to 0' T. 
Since the plan D 1 G E 1 B 1 corresponds to D E of the elevation, or the upper edge 
of the piece— the pattern for which we are about to describe— measurements must 
be made upon the corresponding line in the section, which is G E\ On G E' set off the length of the radii by 
which the "several sections of the plan were struck. Make G'E" equal to S E 1 of the plan, and G D 3 equal 
to C D 1 of the plan, and G E' equal to P P' of the plan. Since the flare of the base, or Xo. 3, is to be equal 
throughout its extent, the several radii as seen in the section will be parallel. Therefore, from the points 





Fig. 352 —Pattern of Front Part (No. »). 
Hip Bath. 



to 



straight 



Fig. 353.— Diagram of Cones for 
Radii of Pattern of Foot. 

Hip Bath. 



112 



Pattern Problems. 



E 3 , D 3 , E 4 draw lines cutting the line O 1 N at angles corresponding to the flare of the piece, as indicated by 
E N or D O of the elevation. Produce these lines in the opposite direction until they meet the vertical line 
O 1 T in the points V, U and T respectively. Then Y E 3 is a radius of that part of the pattern the plan of which 
is shown by E E' E 1 , and II D 3 is the radius of that part of the pattern shown in plan by G D 1 B 1 , and T E 4 is 
the radius of those portions of the pattern shown in plan by G B and B 1 E 1 . In describing the pattern it is im- 
material with which point we commence, since in the plan but one-half has been divided. We will start at the 
point D. Therefore, from any convenient center, as U 1 in Fig. 354, with radius equal to U ~D% describe the arc 
D 2 G 2 , upon which set off the stretchout of D 1 G in the plan. In like manner, from the same center, with the 
same radius, describe the arc D' J B 8 , which make of corresponding length. From G", through the center TJ 1 , 
draw the line G 2 T 2 indefinitely. Set the dividers to the radius T E 4 of the diagram, and measure from G 2 along 
the line G 2 T\ Establish the center T 2 , from which strike the arc G 2 E 2 , which in length make equal to the 
stretchout of G E of the plan. In like manner, from B 8 , also through the center TJ 1 , draw the line B 8 T 1 , 
and with the dividers set as just described, measuring from B 8 , establish the center T 1 , from which describe the 
arc B" E 4 . From E draw a line to the center T 2 , as shown. Set the dividers to V E 3 of the section for radius, 
and, measuring from E 2 , establish the center V, from which describe the arc E 2 E 3 , in length equal to the stretch- 
out of the plan from E to E'. In like manner, using the centers thus established and using lengthened radii of 
the diagram, or, in other words, setting the dividers from the points Y, TJ and T respectively, to the lower line 
of the diagram, Fig. 353, describe the outer line of the pattern, determining the length of the several arcs in 
it by the lines drawn from the several centers produced, all as shown. Then E 3 r* r E 4 will be the pattern of 
the foot of the article, or No. 3 of the elevation. 

500. Patterns of a Coal Hod. — In Fig. 355 we show by elevation and section an ordinary funnel coal hod, 

to be constructed in four pieces. The pieces composing the front 
of the funnel are to be seamed together on the line S E, the back 
and the front are to be joined on the line E Q, and there is to 
be a seam between the base or rim and the body of the hod. The 
inner line of the plan a 2 8 x represents the bottom of the hod or 
the toj> of the foot, as indicated by a 1 8 1 in the elevation. The outer 
line of the plan A B 1 C D E shows the shape of the hod at a 
point through the upper portion corresponding to T X in the ele- 
vation. A B 2 s n shows the shape of the spout, while O 1 1 y 
represents an imaginary section taken through piece No. 3, and is 
introduced to better show correspondence between parts. By trac- 
ing the course of the dotted lines connecting the several portions 
the reader will understand the relationship between the representa- 
tions in plan and the corresponding points in elevation. For the 
patterns we will commence with piece No. 1 in the elevation, or 
the back of the hod. Subdivide the inner line of the plan cor- 
responding to the bottom of this piece, as shown by the small 
figures 1, 2, 3, 4, etc., and from these points carry lines vertically, 
cutting the base of the hod in the points 1', 2 1 , 3 1 , etc. Produce the 
side X S 1 of the piece No. 1 in elevation until it meets the center line in the point II. Then H is the apex of a 
cone, of the envelope of which the required piece is a part. Through the points already obtained in the bottom 
line of the piece, as above described, draw lines from the apex H, producing them until they cut the upper line 
E X of the piece, all as indicated by the dotted lines and by the points I 2 , 2 2 , 3 2 , etc. The axis of the cone, of 
the envelope of which piece No. 1 is a part, is represented by I H. Place the T-square at right angles to the axis, 
and, bringing it against the points in the curved line E X, cut the axis, as shown by the points 3 3 , 4 3 , 5 3 etc. 
From the measurements obtained by these several steps the pattern for the piece No. 1 may be described. To 
avoid confusion of lines, H I, with all the points above described, is transferred to Fig. 356, as shown by H 1 1 1 . At 
right angles to H 1 I 1 , from the point 2 2 , draw a straight line, as indicated, in which set off points equal to the 
distances L 1, L 2, L 3, L 4, etc., in the plan, and through these points, from H 1 , draw lines, as shown, which 
produce indefinitely. With the T-square at right angles to H 1 I 1 , and brought successively against the several 
points at the top of the line, as indicated by the small figures l 3 , 2 a , 3 3 , etc., draw lines, producing them until 
they meet the radial lines just mentioned of corresponding numbers. From IP as center, with radii corre- 




Fig. 354.— Pattern of Foot (No. 3). 
Hip Bath. 



Pattern Problems. 



113 



sponding to the points of intersection thus obtained, describe the arcs, as shown. In like manner, from the 
same center, with radii corresponding to the intersections with the line 2 2 , describe similar arcs. The stretch- 
out of the piece, the pattern of which we are describing, is obtained from the plan, as indicated by the portion 
1 8 y. Take the space in the dividers used in stepping off the plan and, starting with the arc drawn from the 
intersection of the line 2 2 with the line corresponding to 1 of the plan, step to the next arc, and thence 




Fig. 355— Patterns of a Coal Hod.— Elevation, Plan at Line of Foot, and Section near Top. 
step to the third arc. In like manner transfer the entire stretchout of the plan to the pattern, stepping from 
arc to arc. Then a line traced through these points, as shown in the engraving, will be the shape of the bottom 
of the pattern. From H 1 draw radial lines through these points of intersection, producing them until they 
intersect the larger arcs. Then a line traced through the several points of intersection thus obtained, which, as 
indicated in the drawing, terminates at E 3 , will be the shape of the upper part of the pattern. The patterns 



114 



Pattern. Problems. 



of the remaining pieces are obtained, in the main, in the same general manner, and the steps for each are clearly 
indicated in the engravings. It is not necessary, therefore, to give a detailed demonstration of each. There 
are a few points differing, however, from the pattern just explained to which we will call attention. In the 

foot, or flange (No. 2 in the elevation), the same 
points are used as described in connection with 
No. 1. The elevation of No. 2 has been drawn 
to one side from the elevation of the complete 
hod, in order to avoid confusion of lines. The 
various parts or points appear in it as though it 
occupied its normal position. The pattern is 
given in Fig. 357. K 2 M 1 corresponds in all 
respects with K 1 M of the elevation. Inasmuch 
as both boundary lines in the elevation are straight 
lines, this pattern is simpler than that of No. 1, 
of which but one line was straight. Therefore 
all the points of distance from the axis in each 
appear in one line, as they did in the lower line 
of piece No. 1. Thus the points in the line U 
of the elevation appear upon IP of the diagram, 
and all the points on the line M in the elevation 
appear upon the line ~W of the diagram. Dis- 
tances taken from the plan L 1, L 2, L 3, etc., 
are set off on U l , as shown, and lines drawn from 
K 2 through these points cut the line M 1 , giving 
the requisite points in it. All other steps belong- 
ing to this pattern are identical with those of No. 
1. In piece No. 3 a new condition arises. By 
inspection of it in the elevation, it will be seen 
that the profile or section presented in the plan 
is not taken at right angles to its axis, F Q ; therefore the first requirement is to ascertain the shape of the pro- 
file which will fit it when placed at right angles to the axis, as, for instance, on the line cC a\ To obtain this 
we proceed as follows : At any point in convenient proximity to the base of the cone, draw a line at right 
angles to the axis, as O P, upon which to construct the new profile. Subdivide the plan in the usual manner, 
as indicated by the points a, h, c, etc. From these points draw vertical 
lines, cutting the base in the elevation, as shown by a', i\ c\ etc. From 
these points, parallel to the axis of the cone F Q, draw lines cutting O P, as 
indicated by a 3 , i% c% etc. By this means we have the subdivisions in O P 
corresponding to the divisions in O 1 P 1 of the plan. Therefore, to complete 
the profile, set off on each of the lines drawn through O P, measuring to 
each side of it, the distance from corresponding points in O 1 P 1 in the plan 
to the circumference, and trace a line through these points. Having thus 
obtained a section of the cone at right angles to its axis, the remaining steps 
connected with describing piece No. 3, so far as concerns its conical part, are 
identical with those described in connection with piece No. 1. From the 
points a', b\ d, d\ etc., in the bottom line of the piece draw lines at right 
angles to the axis, obtaining the points indicated by a?, I", c", etc. From 
the same points in the bottom line of the piece carry lines toward the apex 
F, cutting the upper line of the piece T U, and from the points thus 
obtained in T U draw lines in like manner at right angles to the axis. The 
line F Q then is transferred with all its points to Fig. 358, as indicated by F 2 Q 2 , and is there used in exactly 
the same manner as the corresponding line in the pattern of the piece No. 1. For the pattern of the portion 
of No. 3 which is flat, and which is outside of the shape derived from the cone, Ave proceed as follows : By 
describing that portion of the pattern derived from the cone, we have obtained the point indicated in the 




356.— Pattern of Back Part of Body (No. i). 
Coal Hod. 




Fig. 357.- 



Pattern of One-harf of Foot (No. 2). 
Coal Hod. 



Pattern Problems. 



115 



engraving by 2 1 . By inspection of the elevation it will be seen that ending at the point 2 1 will be one bonndary 
line of the inner surface. This line in length will be equal to 2 2 IT of piece No. 1. Therefore, with the 
dividers set to this distance as a radius, and with one leg at the point 2 1 as center, describe the arc u r. It then 
remains to find the point in this arc at which the remaining boundary 
line of the pattern will intersect. By inspection of the elevation it 
will be noticed that the division line between the conical part of piece 
No. 3 and the plane surface, or, in other words the line F Q, does 
not intersect with G 1 R within the boundaries of the elevation. There- 
fore it will be noticed that a small portion of the boundary line of the 
pattern lying between the end of the division line between pieces Nos. 
3 and 4 and the cone part of piece No. 3, must be established. S in 
the elevation represents the intersection of the lines F Q and the line 
G 1 R. Upon the line F 2 Q 2 in the pattern establish a corresponding 
point, as indicated by S l . From this point draw a line at right angles 
to F 2 0/ until it meets the corresponding distance line already drawn. 
Then from F 2 , with radius corresponding to this point of intersection, 
describe an arc in a similar manner to the arcs already drawn. The 
point S 2 , at which this arc intersects the stretchout line F 2 l 3 , will be 
the point in the pattern corresponding to S in the elevation, and 
therefore serves as a center from which to measure in order to obtain 
the point sought in the arc u v, to which a line from it is to be drawn. 
The next step, therefore, is to obtain the radius by which to strike the 
intersecting arc. By inspection of the elevation it will be seen that 
this radius will be ecpial in length to the actual distance from S to R. 
From the fact that the line S R lies in a plane which is neither parallel 
to the general plane of the elevation nor parallel to the horizontal line 
drawn through the elevation, it follows not only that we cannot use 
the distance from R to S in the elevation, but that a special opera- 
tion must be per- 
formed in order to 
obtain the actual 
distance indicated. 
This operation con- 
sists of dropping on to the line of the plan G s a point 
corresponding to the point S in the elevation, as shown by 
S 2 in the plan. From R in the elevation draw the horizon- 
tal line R Y. From the intersection of this line with S S 2 
take the distance to S, and set it off from S 2 at right angles 
to the line of the plan G s, all as shown by S 2 V 2 . Through 
the points thus obtained draw the lines s V 2 . Then s V 2 
is the actual length of the line S R in the elevation, and is 
the distance to be used in Fig. 358 as radius by which to 
describe an arc intersecting the arc u v in the pattern. 
Therefore, with S 2 as center, and with radius S V 2 of the 
plan, describe the arc w z, cutting the arc u v in the point 
R 1 . Connect the points S 2 and 2 1 with this point of inter- 
section, R 1 , by straight lines, as shown. Inasmuch as the 
point S in the elevation, from which we have worked to 
obtain the measurement just used, lies outside of the piece No. 3, the boundary line of the pattern, represent- 
ing that portion of the line T IT of the elevation between the line bounding the conical part and the division 
line between it and No. 4, will fall somewhat below S 2 in the pattern, and is to be traced along points corre- 
sponding to the line T U in the elevation, the means for doing which, in full-size work, will be easy to perceive, 
but to indicate which becomes somewhat difficult in a diagram of a scale so small as the accompanying engrav- 





Fig. 35 8.— Pattern of One-half of Front (No. 3). 
Coal Hod. 



Fig- 359- 



-Pattern of Hood (No. 4) 
Coal Hod. 



116 



Pattern Problems. 




Fig. 360.- 



-The Pattern of a Round Pipe to Fit Against a Roof of 
One Inclination. 



ings. In the development of the pattern for piece No. 4 there is nothing different from the steps explained in 
connection with one or the other of the preceding. The profile corresponding to the end S U is assumed to be 
a semicircle, all as indicated by the dotted lines. Connect S U with the section. From this sectional view a 

profile of the piece, as it would appear if 
cnt through the point U at right angles to 
the axis of the cone, is to be constructed, 
all as shown in the second profile, connected 
with S U in like manner by dotted lines. 
Through the points in S IT, from the apex, 
lines are drawn, which are produced until 
they cut the opposite end of the pattern, 
as shown by the points f g, A, etc. From 
these points lines are drawn at right angles to 
the axis. In developing the pattern the line 
G 1 Ft with all its points is transferred to Fig. 
359, as shown by G 2 B 3 . These points are 
used in describing the pattern in the same 
manner as exj)lained in connection with the 
preceding pieces. Respecting the engravings 
here presented, it is to be remarked that 
working to so small a scale as is necessary in 
a book of this character, some inaccuracies in 
proportions, etc., are altogether unavoidable. 
In the plan the inner and outer lines, in actual 
work, would not be parallel. The distance 
between them, measured upon the line 2 C, 
would be less than measured upon the line 
8 D, because the point 2 2 is nearer the apex than S 2 , and also because, the plan being elliptical, the flare is not so 
great laterally as longitudinally. The difference is so slight, however, as to be somewhat difficult to indicate in 
an engraving of small scale. The process employed and described in connection with the patterns, however, 
will be found accurate, and we think it has been presented in such a manner as to enable any one to lay out full- 
size work without doubt or difficulty. 

501. The Pattern of a Pound Pipe to Fit Against a Poof of One Inclination. — In Fig. 360, let A B be 
the pitch of the roof and CFDE the profile of the pipe which is to miter 

against it. Let G O P II be the elevation of the pipe as it is required to be. . 4 s ? 7 1 

Draw the profile in line with the elevation, as shown by C F D E, and. divide it 

into any convenient number of equal parts. Place the T-square parallel to the 

sides of the pipe, and, bringing it successively against the divisions of the profile, 

cut the pitch line, as shown by A P. Lay off a stretchout in the usual manner, 

at right angles to and opposite 

the end of the pipe, as shown by " '° 

I K, and draw the measuring 

lines. Reverse the T-square, 

placing it at right angles to the 

pipe, and, bringing it successively 

against the points in A B, cut 

the corresponding measuring 

lines. A line traced through the 

points thus obtained, as shown by 

L M N, will finish the pattern. Fi C- 36t.— The Pattern of an Elliptical Pipe to Fit Against a Roof of One Inclination. 

502. The Pattern of an Elliptical Pipe to Fit Against a Poof of One Inclination.— In Fig. 361, let 
N C D O be the elevation of an elliptical pipe fitting against a roof, represented by A P. Let E FG be the 
section or profile of the pipe. Draw the profile in convenient proximity to the elevation, as shown, and divide it 




Pattern Problems. 



117 



into any convenient number of equal parts. Place the T-square parallel to the sides of the pipe, and, bringing 
it against the points in the profile, drop lines cutting the roof line A B, as shown. Opposite to the end of the pipe, 
and at right angles to it, lay off a stretchout, as shown by H I, and through 
the points in it draw measuring lines in the usual manner. Reverse the 
T-square, placing it at right angles to the pipe, and, bringing it successively 
against the points in A B, cut the corresponding measuring lines, as indi- 
cated. A line traced through these points, as shown by K L M, will be the 
required pattern. In the illustration the long diameter of the ellipse, or 
E G, is shown as crossing the roof. The same rule applies if the pipe is 
placed in the opposite position — that is, with Q F crossing the roof — the 
only change required being in the position of the profile, which, of course, 
would require to be turned around, and drawing the elevation to correspond 





503. 



Fig. 362. — The Pattern 0/ an Octagon Shaft Fitting Over the Ridge of a Roof. 
with it. Otherwise proceed in all respects as above. From this it is evident that a pattern for the pipe, when 
its section lies diagonally, may be described by the same rule. 

The Pattern of an Octagon Shaft Fitting Over the Ridge of a Poof. —In Fig. 362, let AB C be the 
section and D H G I E the elevation of an octagon shaft inhering against a roof, repre- 
sented by the lines F G and G K. Draw the section in line with the elevation, as shown, 
and from the angles drop lines, giving T V and U W of the elevation. Drop the point 
G back on to the section, thus locating the points 9 and 4. Opposite the end of the shaft, 
and at right angles to it, draw a stretchout line, as shown by S R, and through the points 

in it draw measuring lines in the 
3456789io98 75543'),, usual manner. Place the T-square 

at right angles to the shaft, and, 
bringing it successively against the 
points in the roof line formed by 
the intei'section with it of the angle 
lines in the elevation, and also 
against the point G, representing 
the ridge of the roof, cut the corre- 
sponding measuring lines. Then a 
line traced through the points thus 




Fig. 363.— The Pattern of a Round Pipe to Fit Over the Ridge of a Roof. 

obtained, all as shown by P O N M L in the engraving, will be the pattern required 
504. The Pattern of a Pound Pipe to Fit Over the Ridge of a Roof. 



Let A B C in Fig. 



363 be a sec- 



118 



Pattern Problems. 



tion of the roof and D S B T E an elevation of the pipe. Draw a profile of the pipe in line, as shown by 
F G II. Since both inclinations of the roof are to the same angle, both halves of the pattern will be the same. 
Therefore space off but .one-half of the profile for dropping the points on to the roof line. Lay off a 
stretchout, however, equal to the whole profile, numbering the points in both halves correspondingly. Draw 
measuring lines through these points in the usual manner. Place the T-square parallel to the sides of the pipe, 
and, bringing it against the points in the profile, cut the roof line, as shown from B to T. Beverse the T-square, 
placing it at right angles to the lines of the pipe, and, bringing it successively against the points dropped upon 
the roof line, cut the corresponding measuring lines. A line traced through the points, as shown byLMNOP, 
will form the required pattern. 

505. The Patterns (fa Cylinder Uttering over the Peak of a Gable Coping having a Double Wash. — Let 
ABC in Fig. 364 be the elevation of a coping to surmount a gable, the profile of which is D E FE 1 D 1 , which, 
as will be seen, shows a double wash, E F and F E'. Let IOPN be the elevation of a pijse or shaft which 
is required to miter over this double wash at the peak of the gable. For the pattern proceed as follows: In 
line with the pipe or shaft construct a profile of the same, as shown by G 1 L 1 K? H 1 , which divide into any 
convenient number of equal parts, and from the points thus obtained drop lines vertically on to the elevation. 




Fig. 364.— The Patterns of a Cylinder Mitering over the Peak of a Gable Coping Having a Double Wash. 

Draw a corresponding profile, as shown by II G L K, directly over the profile of the coping, all as shown, 
which divide into the same number of equal parts, beginning at a corresponding place in the profile, and 
from the points in it drop lines on to the profile of the coping, cutting the washes E F and F E 1 , and 
thence carry the lines parallel to the lines of the coping, producing them until they intersect the lines dropped 
from the profile G 1 L' E? LP. Through the points of intersection thus obtained trace a line, as shown from O 
to P, then OB'P will be the miter line in elevation. In line with the end M N of the shaft, and .at right 
angles to it, lay off a stretchout of the profile G" IF K 1 L', as shown by B S, in the the usual manner, through 
the points in which draw measuring lines. Commence numbering these measuring lines with the figure corre- 
sponding to the point at which the seam is desired to be, in this case 5. Place the T-square at right angles 



\l 



Pattern Problems. 



119 



to the shaft, and, bringing it against the points in the miter line O B 1 P, cut the corresponding measuring 
lines. Then a line traced through these points of intersection, as shown by T U Y ¥ X, will be the pattern 
required. In case it should be desired to miter the coping against the base of the shaft, the pattern for it may 
be obtained from the same lines in the following manner : At right angles to the lines of one side of the 
coping, as A B, lay off a stretchout of the wash of the coping, E F E 1 , all as shown by E 1 F 1 E 3 . In this stretch- 
out line set off points corresponding to the points in E F E', obtained by the lines previously dropped from 
the profile G II K L. Place the T-square at right angles to A B, and, bringing it against the points in the 
miter line B 1 , cut lines of corresponding 
numbers drawn through the stretchout E" E 3 , 
all as indicated by the dotted lines. Then a 
line traced through these points of intersection, 
as shown by Z I Y, will be the pattern of the 
wash required to miter against the base of the 
shaft. In case the shaft is octagonal in shape, 
the same general rules apply. Less divisions, 
however, will be required in the profile, it 
only being necessary to drop points from the 
angles, being, in this respect, identical with 
Section 503. 

506. The Pattern of a Flange to Fit 
Around a Pipe and Against a Poof of One 
Inclination. — Let L M, Fig. 365, be the incli- 
nation of the roof and PETS an elevation 
of the pipe passing through it. N then 
represents the length of the opening which is 
to be cut in the flange, the width of which 
will be the same as the diameter of the pipe. 
Let A B D C be the size of the flange desired, 
as it would appear if viewed from a point 
directly above the pipe. Immediately in line 
with the pipe draw the profile GHIK, put- 
ting it in the center of the plan of the flange 
ABDC, or otherwise, as required. Divide 
one-half of the profile in the usual manner, 
and carry lines vertically to the line L M, rep- 
resenting the pitch of the roof, and thence, 
at right angles to it, indefinitely. Carry 
points in the same manner from A and B. 
Draw C 1 D 1 parallel to L M. Make C 1 A 1 
equal to A C, or the width of the required 
flange, and draw A 1 B' parallel to C D 1 . Then 
C A 1 B 1 D 1 will be the pattern of the required 
flange. Draw E 1 F' through it at a point cor- 
responding to E F of the plan, crossing the 
lines drawn from the profile. From E 1 F 1 set 
off on each side, on each of the measuring lines crossing it, the width on corresponding lines, measuring from E F 
in the plan to the profile. Through the points thus obtained draw a line, which will give the shape of the 
opening to be cut, all as shown by G 1 II 1 I 1 K 1 . 

507. A Conical Flange to Fit Around a Pijye and Against a Poof of One Inclination. — In Fig. 366, 
is shown, by means of elevation and plan, the general requirements of the problem. A B represents the pitch 
of the roof, GHKI represents the pipe passing through it, and C D F E the required flange fitting around 
the pipe at the line C D and against the roof at the line E F. The flange, as we have drawn it, becomes a sec- 
tion of the envelope of a right cone. By prolonging E C and F D until they intersect at ~W, the apex is found, 




365. — The Pattern of a Flange to Fit Around a Pipe and Against a 
Poof of One Inclination. 



120 



Pattern Problems. 



and by continuing these same lines in the opposite direction, to L and M respectively, and drawing the line 
L M, a section of the cone is described, from the envelope of which the flange is cut. In connection with the 
elevation just described, we have shown a plan of the several parts, or a representation of them as they would 

appear if viewed from above. S T represents the pipe and X O the 
the flange. While the pipe is made to pass through the center of the 
cone, as may he seen by examining the base line L M in the elevation, 
and- also P R of the plan, it does not pass through the center of the 
oblique cut E F in the elevation, or, what is the same, NO of the plan. 
For the pattern of the flange proceed as shown in Fig. 367, which in the 
lettering of its parts is made to correspond with Fig. 366, just described. 
Divide the plan PIE into any convenient number of parts — in this 
case twelve — and from each of the points thus established erect perpen- 
diculars to the base of the cone, obtaining the points l 1 , 2 1 , 3 l , etc. From 
these points draw lines to the apex of the cone W, cutting the oblique 
line E F and the top of the flange C D, as shown. Inasmuch as C D 
cnts the cone at right angles to its axis, the line in the pattern correspond- 
ing to it will be an arc of a circle ; but with E F, which cuts the cone 
obliquely to its axis, the case is different. A measurement in the pattern 
is required at each point, corresponding to the divisions given in the plan. 
Accordingly, the several points in E F, obtained by the lines from the 
plan drawn to the apex W, must r 1 

be transferred to one of the sides 
of the cone. From the points 
3 , I s , 2% 3 3 , in E F, draw lines at 
right angles to the axis of the 
cone W X, cutting the side W M, 
as shown. We now have all the 
points necessary to use in describ- 
ing the pattern. 
With W as center, 
and with W M as 
radius, strike the arc P 1 It 1 indefinitely, and, with the 
same center and with W D as radius, strike the arc C 1 D 1 
indefinitely, which will form the boundary of the pattern 
at the top. At any convenient distance from W M draw 
W P 1 , a portion of the length of which will form the 
boundary of one end of the pattern. On P' P 1 , com- 
mencing with P 1 , set off spaces equal in length and the 
same in number as the divisions in the plan P X P, all 
as shown by a , T, 2", 3 3 , etc. From these points draw 
lines to the center W, as shown. With one point of the 
dividers set at W and the other brought successively to 
the points cut in W M by the horizontal lines drawn 
from E F, cut the corresponding lines in the stretchout 
of the pattern, as indicated by the curved dotted lines. 
A line traced through these points, as E' F 1 , will repre- 
sent the lower side of the pattern. As we used hut one- 
half of the plan in laying out the stretchout, the pattern 
C 1 E 1 F 1 D 1 thus obtained is but one-half of the piece 
required. In use it is to be doubled. The seam can be 
made to come through the short side at C E 
508. The Pattern of a Flange to 




Fig. 366.— Elevation and Plan. 
A Conical Flange to Fit Around a Pipe 
■ and Against a Roof of One Inclination. 




Fig. 367.— Pattern. 

A Conical Flange to Fit Around a Pipe and Against a 

Roof of One Inclination. 



or through the long side at D F, at pleasure. 
Fit Around a Pipe and Over the Ridge of a Roof. — In Fig. 368, let 
A B B C be the section of the roof against which the flange is to fit, and let O P S P be the elevation of the 



Pattern Proble 



'ins. 



121 



pipe required to pass through the flange. Let the flange in size be required to extend from A to C over the 
ridge B. By inspection it will be seen that the process of describing the pattern is identical with that in Sec- 
tion 500. Produce C B, as shown by B A', making B A 1 equal to B A. Proceed as in the manner described 
in the problem just referred to. Divide the profile 
DEF6 into any number of equal parts in the usual 
manner, and from the points so obtained carry lines 
vertically to the line A 1 C, and thence, at right 
angles to it, indefinitely. Also carry lines in a similar 
manner from the points A 1 and C. Draw II L. Hake 
II I the width of the required flange, and draw I K 
parallel to II L. Connect K L. Through that part 
of the flange in which the center of the required open- 
ing is desired to be, draw the line A" C, crossing the 
lines drawn from the profile. From each side of this 
line, on the several measuring lines, set off the same 
distance as shown upon the corresponding lines between 
D F of the profile and the circumference. A lino 
traced through the points thus obtained, as shown by 
D' E 1 F 1 G 1 , will be the required opening to fit the 
pipe. Through the center, across the flange, draw the 
line N M, which represents the line of bend corre- 
sponding to the ridge B of the section of the roof. 





Fig. 368.— The Pattern of a Flange to Fit Around a Pipe and 
Over the Ridge of a Poof. 

509. A Two-Piece Elbow.— -In Fig. 369, let A G B D be the pro- 
file of the pipe in which the elbow is to be made. Draw an elevation of 
the elbow as it is required to be, as shown by E 6 I H K F. Draw the 
diagonal line G K, which represents the joint to be made. Draw the 
profile of the pipe in line with one arm of the elbow, as shown. Divide 
the profile into any convenient number of equal parts. Place the T-square 
parallel to the lines of the arm of the elbow, opposite the end of which 
the profile has been drawn, and, bringing the blade successively against 
the several points in the profile, drop corresponding points on the miter 
or joint line K G, as shown by the dotted lines. Opposite the end of the 
same arm, and at right angles to it, lay off a stretchout line, M 1ST, divided 
in the usual manner, and through the divisions draw measuring lines, 
as shown. Place the blade of the T-square at right angles to the same 
arm of the elbow, or, what is the same, parallel to the stretchout line, 
and, bringing it successively against the points in K G, cut the corre- 

as shown. A line traced through these points, 



ring lines, 



Fig. 369. — .4 Two-Piece Elbow. 

spondm s 

as indicated by E P O, will form the required pattern, 

510. A Three-Piece Elbow.— -In Fig. 370, letEMLIIIKNF be the elevation of a three-piece elbow. 

Draw the profile ABC in line with one arm, as shown, and divide it into any convenient number of equal 

parts. Draw the joint or miter lines M IN" and L K. Place the blade of the T-square parallel to the arm of the 

elbow opposite the end of which the profile has been drawn, and, bringing it against the points in the profile, 



122 



Pattern Problems. 



drop corresponding points upon the miter line M K Shift the movable head of the T-square, so that the blade 
lies parallel to the second section of the elbow, or the same thing may be accomplished by using the 45-degree 

set-square, and, bringing it against the 
points in M N", drop like divisions upon 
L K. At right angles to the second sec- 
tion, lay off a stretchout of the profile 
ABC, as shown by P 0, through the 
points in which draw measuring lines in 
the usual manner. Placing the T-square 
so that the blade shall come at right 
angles to this section, or, what is the 
same, parallel to the stretchout line, bring 
it successively against the several points 
in the miter lines M 1ST and L K, and cut 
the corresponding measuring lines. Then 
lines traced through these points, as 
shown by D X T and G "W Z, will be 
the pattern of the middle section. For 
the end sections of the elbow proceed as 
follows : Opposite the end of and at 
to the arm draw a stretch- 
out, as R S, through the divisions in 
which draw measuring lines in the usual 
manner. Placing the T-square at right 
to the arm, and bringing it suc- 




Fig. 370. — A Three-Piece Elbow. 

cessively against the points in K L, cnt the 
corresponding measuring lines, as shown. 
Then the line T U V, traced through 
the points thus obtained, forms the pattern 
of an end section. 

511. A Four-Piece Elbow. — To draw 
the elevation of a four-piece elbow proceed 
as follows : Lay off the arms E6FD and 

to 



371- — -4 Four-Piece Elbow. 



angles 



N I L M in Fig. 371 at right 

each other, and draw the diagonal line a d, upon which they would intersect if produced indefinitely. Estab- 
lish the point a on this diagonal line at convenience, and from it draw the lines a I and a at right angles to 



Pattern Problems. 



123 



the two arms of the elbow respectively. Draw c d and b d, thus completing the square ab d c. From a as 
center, and with a b as radius, describe the arc bfee, as shown, which divide into three equal parts, thus obtain- 
ing the points / and e. Through / and e, to the center a, draw the lines / a and e a, which will represent the 
centers of the middle sections of the elbow, at right angles to which the sides of the same are to be drawn. 
Through f, and at right angles to f a, draw L K, meeting M L in the point L, and stopping on the line a d at 
the point K. Through e, and at right angles to e a, draw a line, commencing in the point K and terminating 
in G where it meets the line E G. In like manner draw the lines of the inner side of the elbow, as shown by 
F H and H I. Draw the miter or joint lines F G, II K and L I, as shown. For the patterns proceed as fol- 
lows : In line with one arm of the elbow draw 
a profile, as shown by ABO, which divide into 
any convenient number of equal parts. Place 
the T-square parallel to this arm of the elbow, 
and, bringing the blade against the points in the 
profile, drop corresponding points upon the miter 
line F G. Change the T-square so that its blade 
shall be parallel to the lines of the second section 
of the elbow, and, bringing it against the points 
in F G, cut corresponding points on II K. Op- 
posite the end of and at right angles to the lower 
arm of the elbow, lay off the stretchout line 
O P, as shown, through the divisions in which 
draw the usual measuring lines. Place the 
T-square at right angles to the arm of the elbow, 
and, bringing it successively against the points 
in the miter line F G, cut the corresponding 
measuring lines. Then a line traced through 
the points thus obtained, as shown from P to T, 
will be the pattern of one of the arms. Produce 



a e, representing the middle of the second section 
in the elbow, as shown by V "W", upon which lay 
off a stretchout, and through the points in the 
same draw measuring lines. Placing the T-square 



parallel to 



a e, 



or, what is the same, at • right 




Fig. 372. — A Five-Piece Elbow. 



angles to the section in the elbow, bring it against 
the several points in the miter lines II K and 
F G, and cut the corresponding measuring lines. 
Then lines traced through the points thus obtained, 
as shown from X to Z and Y to S, will give the 
pattern. 

512. A Five-Piece ETbow. —The elevation of 
a five-piece elbow may be drawn as follows : Lay 
off the two arms (Fig. 372) at right angles to 
each other. Draw the line g a indefinitely, upon 
which they would meet if sufficiently prolonged. Establish the point a in this diagonal line with reference to 
the curve which it is desired the elbow shall have, and from it, at right angles to the two arms of the elbow 
respectively, draw a b and a c. From a as center, with a b as radius, describe the arc b f e c, which divide into 
four equal parts, thus obtaining the points d, e and f, from which draw lines to a, all as shown by d a, e a and 
fa. Then these lines represent center lines of the several sections of which the elbow is composed, and at 
right angles to which the sides are to be drawn. Through f and at right angles to f a, draw Y S, joining the 
side of the arm E S in the point S, and a corresponding line drawn through e in the point Y. In like manner 
draw the line T P, representing the inner side of the same section. The remaining sections are to be obtained 
in the same way. As but one section is necessary for use in cutting the patterns, the others may or may not be 
drawn, all at the option of the pattern cutter. Draw the miter or joint lines S R and Y T. Opposite one arm 



124 



Pattern Problems. 



draw a profile, as shown by BAC, which subdivide in the usual manner. Place the T-square parallel to the 
lines of the arm, and, bringing the blade against the several points in the profile, drop corresponding points 
upon the miter line S E. Shift the T-square so that the blade shall be parallel to the part T S E T, and trans- 
fer the points in S E to V T, as shown. For the pattern of the arm, at right angles to it and opposite the end 
lay off a stretchout, as shown by F G, through the points in which draw the usual measuring lines. Place the 

T-square at right angles to 
^r p the arm, and, bringing it 

■>-"\\ \ \ against the points in E S, 

cut the corresponding meas- 
uring lines, as shown. Then 
a line traced through these 
points, as shown from H to 
I, will be the pattern. For 
the pattern of the sections 
prolong the line a f, as 
shown by L K, upon which 
lay off a stretchout, through 
the points in which draw 
the measuring lines in the 
usual manner. Placing the 
T-square at right angles to 
the section, or, what is the 




same, parallel to the stretchout line, bring it against 
the several points in the lines K S and T V, and cut 
the corresponding measuring lines. Then lines 
traced through the points thus obtained, all as 
shown by N P and M O, will be the pattern 
sought. 

513. Elbow at Any Angle.— Let DFHKLIGE, 
Fig. 373, represent a pipe in which elbows are re- 
quired at odd angles. In drawing the elevation 
care is to be taken that the lines representing the 
sides of the pipe be parallel and the same distance 
apart throughout. In convenient proximity to and 
Fig. ITi.—mow at Any Angle. i n ] me ^fth one eru ] f t h e pipe fl raw a profile, as 

shown by A B C, which divide in the usual manner. Placing the T-square parallel to the first section of the pipe, 
and, bringing it against the several points in the profile, drop corresponding points upon F G. Shift the T-square, 
placing it parallel to the second section, and, bringing it against the several points in F G, drop corre- 
sponding points upon II I. At right angles to the first section, and opposite the end of it, lay off a stretchout 
line, as shown by T U, through the points in which draw the customary measuring lines. Placing the T-square 
at right angles to this section of the pipe, and bringing it against the several points in F G, cut the correspond- 
ing measuring lines. Then the line E S traced through these points will be the other end of the pattern 
sought. The pattern for the opposite end is to be obtained in like manner, all as shown byMK'O P, and 



Pattern Problems. 



125 



therefore need not be described in detail. For the pattern of the middle section proceed as follows: At right 
angles to it lay off a stretchout, W Y, with the customary measuring lines. Placing the T-square at right 
angles to the section, bring it successively against the points in G F and I IT, and cut the corresponding 
measuring lines, as shown. Then lines traced through these points, as shown by T X and Q Z, will be the pat- 
tern sought. The positions of the longitudinal joints in the several sections of this elbow, as well as those of 
all others, are determined by the order in which the measuring lines drawn through the stretchout are num- 
bered. In the present instance we have allowed the joints to come on the back of the pipe, or, in other words 
upon DFHX, which corresponds to the point 1 in the profile. Hence, in numbering the measuring lines in 
the several stretchouts, we have placed 1 at the commencement and ending, while if we had desired the joint to 
come on the opposite side, or at the point corresponding to 9 of the profile, we would have commenced and 
ended with that figure in numbering the measuring lines, the figure 1 in that case in regular order comin°- 
where 9 now occurs. 

514. A Pipe Carried Around a Semicircle oy means of Cross Joints. — In Fig. 374, let F E D be the 
semicircle around which a pipe, of which A C B is a section, is to be carried by means of any suitable number 
of cross joints, in this instance ten. Divide the semicircle FED into the same number of equal parts as 




Fig. 374. — A Pipe Carried Around a Semicircle by means of Cross Joints. 

there are to be joints, which, as just stated, in the present case is ten, all as shown by D, 0, P, Ft, S, E, etc. 
From each of these points, D, 0, P, Ft, etc., draw lines to the center Z, as shown. Obtain points intermediate 
between E S, S B, P P, etc., as shown in the engraving by T, X, V, etc., through which draw lines from the 
center Z indefinitely. Connect the points E S, S P, P P, etc., by drawing lines at right angles to Y Z, X Z, 
Y Z, etc. From D draw D Z. Set off a space, D A 1 , equal to the diameter of the pipe. Draw a semi- 
circle, as indicated by the dotted line, and obtain the inner line of the pipe in the same manner as just 
described for the outer line. Draw the profile of section ABC directly below and in line with one end 
of the pipe, all as shown in the engraving. As may be seen by inspection of the diagram, two patterns are 
required, one corresponding to the half section occurring at the end, and the other corresponding to the full 
sections composing the body of the pipe. The pattern for the latter may be obtained as shown in the engrav- 
ing, or, if preferred, it may be obtained by making a duplicate of one-half of the larger piece. For the 
pattern of the end section proceed as follows : Divide the profile A B C in the usual manner into any con- 
venient number of equal parts, and from the points thus obtained carry lines upward at right angles to Z D, 
cutting T 1 T. Prolong the line Z D, and upon it place a stretchout from the profile A C B. perpendicular to 
which draw measuring lines in the usual manner. "With the T-square placed parallel to Z D, and brought suc- 
cessively against the points in T 1 T, cut the measuring lines of corresponding numbers. Then a line traced 
through the points of intersection thus obtained will be the shape of the pattern sought, all as shown by IKL. 



126 



Pattern Problems. 



Then G I Iv L H will be the complete pattern for one of the end sections. For the pattern of the large sections 
lay off a stretchout opposite the center of any one of them, and upon a line radial from the center, as shown by 

M N, and through the points in it 
draw measuring lines in the usual 
manner. Place the T-square paral- 
lel to the stretchout line, and, bring- 
ing it against the several points in 
the miter lines TJ 1 U and Y 1 V, 
which are obtained by carrying the 
points from T 1 T by lines drawn par- 
allel to each section through which 
they pass, cut the corresponding 
measuring lines, all as shown, thus 
completing the pattern. 

515. To Form a Semicircle in 
a Pi_pe by means of Longitudinal 
Seams. — By the nature of the prob- 
lem the pipe resolves itself, with 

Fig' 375.— Elevation and Section. / . . 

respect to its section or profile, into 
some regular polygon. In the illustration presented in Fig. 375 an 
octagonal form is employed, but any other regular shape may be 
used, and the patterns for it will be cut by the same rule as here 
explained. In Fig. 375, let X L V be some semicircle around which 
an octagonal form is to be carried. Draw 1ST V, passing through the 
center "W. Through "W draw the perpendicular L K indefinitely. In 
convenient proximity to one end of the semicircle construct a profile, 
as shown by A B C D F H G E, letting points in it fall directly below 
corresponding points on the line N V, all as shown in the engraving. 




6\ 



\ 



\ 



\. 



V 



\. 



\ 



\ 



\ 




By inspection of the dia- 
gram it is evident that 
the pattern for the sections 
corresponding to U T P 
in the elevation, may be 
pricked directly from the 
drawing as it is now con- 
structed, and that the pat- 
terns for the sections repre- 
sented by E A and D F of 
the profile, will be plain 
straight strips of the width 
of one side of the figure, 
as shown by either E A 

or D F, and in length cor- N' 0' P 1 B' 

responding to the length of Fi s- 37s.— Pattern. 

the sweep of the elevation ^° ^ r ° rm a Semicircle in a Pipe by means of Longitudinal Seams. 

on the lines 1ST L V and EXS respectively. But for the two sets of pieces, represented by 1ST V TJ O and 
P T S R in the elevation, additional steps are to be taken. Prolong the side H F of the profile until it cuts 
the center line L K of the elevation in the point M. Then M F and M H are the radii of the pieces corre- 



Pattern Problems. 



127 



sponding to P T S E of the elevation. Prolong the side E (1 of the elevation until it cuts the center line in 
the point M 1 . Then M' G and M' E are the radii of the pieces corresponding to 1ST Y U O of the elevation. 
These radii are to be used as shown in Fig. 376. From M 2 in Fig. 376 as center, using each of the several radii 
in turn, strike arcs indefinitely, as shown by N' Y\ O' U 1 , P 1 T' and R' S 1 . Step off the length N" V in the. 
elevation, Fig. 375, and make ISP V of Fig. 376 equal to it. Draw ISP O 1 and V 1 E 1 radial to W. In like 
manner establish the length of P 1 T 1 , and draw P 1 E 1 and T 1 S 1 , also radial to the center, as shown. It is evi- 
dent by inspection of Fig. 376 that if the patterns for the two pieces are struck from a common center, as we 
have shown, it is only necessary to step off the length upon one member. By drawing radial lines, as shown, 
the other arcs will be intercepted at the proper points. This rule may be employed for carrying any polyg- 
onal shape around any curve which is the segment of a circle. The essential points to be observed are the 
placing of the profile in correct relationship to the elevation and to the central line L K. Then prolong the 




M G K c H 

Fig. 377. — Two-Piece Elboiv in Tapering Pipe. 

oblique sides until they cut the central line, thus establishing the radii by which they may be struck. In 
the case of elliptical curves, by resolving them into segments of circles and applying this rule to each section, 
as though it were to be constructed alone and distinct from the others, no difficulty will be met in describing 
patterns by the principles here set forth. The several sections may be united so as to produce a pattern in one 
piece by joining them upon their radial lines. This principle is further explained in the problem of the 
patterns for the curved molding in an elliptical window cap. See Section 569. 

516. Two-Piece Elbow in Tapering Pijpe.^-In Fig. 377 is represented an elbow constructed in two pieces 
occurring in taper pipe. The several steps required for the development of the pattern are as follows : Pro- 
duce the sides A I and B D of the upper piece of the elbow until they meet in the point E. Then E is the 
apex and E B and E A the sides of a cone of which I D B A is a section. Produce the axis E S to any 
convenient point, as Z, through which draw TU at right angles to the axis. Produce the sides I A and D B 
until they meet T U in the points T and U, as shown. The next step is to construct a section of the cone as 
it would appear when cut on the line T TJ. Through any convenient point below the lower section, and at 



128 



Pattern Problems. 



right angles to the axis of the lower section, draw a straight line, as M N. From the points A and B of the 
miter line between the two sections, drop lines parallel to the axis of the lower section, cutting M N in the points 
G and II. From the point C, midway between G and H, as center, with C G or C II as radius, describe the 
semicircle G L H. Then GLH may be regarded as a plan of the miter A B upon a horizontal plane. From 
the extremities P and R of the bottom line of the lower section drop points parallel to its axis, as shown at 
M and N. From the point K, midway between M and N, and also in line with the axis S produced, with 
radius K M or K N, describe the semicircle M IN". Then M N is the plan of the lower section at the base 
P R. Divide HON into any convenient number of equal parts in the usual manner, and from the points thus 
obtained carry lines vertically to the base P R ; also from the same points draw lines to the center K, cutting 
GLH. Produce the sides P A and R B until they meet in the point F. From the points in P R carry lines 
toward the apex F, cutting A B. Produce the axis of the upper section beyond Z to any convenient point, as 

K 1 . Through K', at right angles to the axis T Z produced, draw Y W. Upon 
Y "W set off C, the same distance from K 1 that C is from K. in the line H N. 
From C 1 , with radius equal to that used in describing the profile GLH, describe 
the semicircle G 1 L 1 H', in which set off points corresponding to the divisions 
in GLH. From K 1 , through these points in H L 1 G 1 , draw radial lines indefin- 
itelv, as shown by K' 2, E? 3, etc. From the apex E, through the points in the 
miter line A B, obtained from the plan of the lower section, as already described, 

draw lines cutting T U, as shown by the dotted lines. 
From the points T U draw lines parallel to the axis, 
cutting the radial lines drawn from K 1 through the 
points in the profile H 1 L 1 G 1 . Through the points 
of intersection between lines of corresponding num- 
bers thus obtained trace a line, as shown by V X "W". 
Then V X W is the plan of the cone, of which I D B A 
is a section as it would appear when cut on the line 
T U. From the same points in A B, as already 
described, draw lines at right angles to the axis, cut- 
ting it as shown by the points above and below S. 
From any convenient point, as E' in Fig. 378, draw 
the straight line E 1 Z', which make equal to E Z of 
the elevation. Set off E 1 T 1 equal to E T of the ele- 
vation ; also set off in E 1 Z 1 points corresponding to 
the points in E Z of the elevation, obtained from the 
miter line A B above described. From Z 1 , at right 
angles to E 1 Z 1 , set off Z 1 U 1 , upon which, measuring 
from Z 1 , set off distances corresponding to each of the 
radial lines, measuring from K 1 to the profile V X "W, all as indicated by the small figures. From the points 
thus obtained in Z 1 IT 1 , draw lines to E 1 . Intersect each of these lines by a line drawn from the corresponding 
point in E 1 Z 1 , all as shown in the diagram. From E 1 as center, with radii E'l, E'2, E'3, etc., describe arcs, as 
shown between B s and A 2 , upon which set off the stretchout of the profile V X ~W, by stepping from arc to arc. 
Through these points draw lines to the center E 1 . From E 1 as center, with radii corresponding to the several 
intersections between the Hues drawn from the points in E' Z 1 , and the lines drawn from E 1 to the points in 
Z' U'j describe arcs which shall intersect lines of corresponding numbers drawn from the arc B 2 A" to the center 
E 1 . Inasmuch as the point Y 1 , corresponding to Y of the elevation, represents all these points, each of these 
lines will be intersected by an arc corresponding to the intersection of the line Y 1 and the line drawn from E 1 
to one of the points in Z 1 IP, all as shown in the diagram. Then lines traced through the intersections thus ob- 
tained, as shown from D' to I 1 , and also from B 1 to A 1 , will be the top and bottom lines of the required pattern. 
Connect D 1 B 1 and I 1 A 1 . Then D 1 I 1 A 1 B' is the half pattern of the upper section. For the pattern of the 
lower section, from any center, as F, Fig. 377, with F P as radius, describe the arc P 1 R', upon which lay off a 
stretchout of the plan M rT, making points in P 1 R' to correspond with the points in the plan. Through 
these points draw lines to the center F, as shown. From points in the miter line A B draw lines at right 
angles to the axis F 0, cutting the side F P, as shown in the points below A. From F as center, with radii 




J234567 3 9 y 

Fig. 378.— Pattern of Upper Section. 
Two-Piem Elbow in Tapering Pipe. 



Pattern Problems. 



129 



corresponding to the several points below A just mentioned, describe arcs, as shown by the dotted lines each 
of which produce until it cuts the line bearing a corresponding number drawn from the arc P' B 1 to the center 
F. A line traced through these points, as shown by A 3 B 3 , will be the upper line of the required pattern of the 
lower section, and A 3 P 1 E 1 B 3 will be the half pattern of the lower section. 

517. Three-Piece ETboto in Tajpermg Pipe. — In Fig. 379 is shown a three-piece elbow occurring in taper 
pipe, in which the flare is uniform throughout the three sections. The usual method of constructing the 
patterns for such an elbow would be the same as have been described for the two-piece elbow in the last 
demonstration. A short method, however, is available, both in three-piece and in two-piece elbows. 
Having described the ordinary method as 
applied to a two-piece elbow, the short 
method may be described in connec- 
tion with the three-piece elbow as fol- 
lows : The sections of which the elbow 



tv. <cr; 




MOEFEPNLis composed 
are cut from the 
shown by E G F. As drawn, 
the lower section of the el- 
bow P E F E corresponds 
with the lower section of the 
cone, E F being the base com- 
mon to both. The second sec- 
tion of the elbow E P ¥ 
corresponds with O'EP N 1 , 
and the third section of the 
elbow M 1ST L corresponds 
with M 1 O 1 N 1 L 1 of the cone. 
The principle upon which the 
patterns are cut is that by 
which the envelope of any 
section of the cone is described 



Elevation of Elbow and Cone from 
which Sections are taken. 



Three-Piece Elbow in Tapering Pipe. 



The essential point requiring attention, therefore, is the means by which the 
lines P E and W 0\ which divide the cone into sections, shall be located so that the several sections of the 
cone shall, when joined together, constitute the elbow that is required. To find the angle of the miter line, or 
the line of cut through the cone, lay off the angle of the elbow, as A B C. Bisect this angle by the line D B. 
Then D B represents the direction across the cone at which the cut must be made. Having thus obtained the 
direction of the line, at any required bight draw P E parallel to D B. In like manner, for the second section, at 
any convenient point against the axis lay off the angle SHI, corresponding to the angle desired between the 
second and third sections. Bisect this angle, as shown by H T. From H as center, with any convenient radius, 
describe the arc O 2 T W. Hpon either side of the cone, according to convenience, locate a point representing 
the length of one of the sides of the second section, as, for example, O'. Set the dividers to O 3 T, and from 



130 



Pattern Problems. 






O 1 as center, with this radius, cut the axis of the cone in the point T". From T as center, with radius T W, 
cut the side of the cone in N 1 . Draw IN"' O 1 , which will be the line of cut dividing the second and third sec- 
tions. A simpler method of obtaining these lines is as follows : The lower section of the elbow E F E T cor- 
responds with the cone already. At the required hight draw P E. Upon the sides of the cone set off E O 1 
equal to P K Set off P K" 1 equal to E 0. Then draw ISP O 1 as before. Having thus obtained the lines of 
cut through the cone, the patterns may be described as follows : Draw the plan V W Y, its center X falling 
upon the axis of the cone produced, which divide in the usual manner into any convenient number of equal 
parts. Through the points thus obtained erect perpendiculars to the base E F, and thence carry them 
toward the apex G, cutting the miter lines P E and 1ST 1 O 1 . With the T-square at right angles to the axis G C, 
and brought successively against the points in W 0' and P E, cut the side G F of the cone, as shown by the 
points above O 1 and below E. From G as center, with radius G F, describe the arc E 1 F 1 , upon which lay off 
the stretchout of the plan V W Y, as shown by the small figures 1, 2, 3, etc., and from these points draw lines to 

the center G. From G as center, 
describe arcs corresponding to the 
several points established in G F 
from the miter lines already de- 
scribed, which produce until they 
intersect lines of corresponding 
numbers drawn from the center G 



to the arc E 1 F 1 . Through these 
points of intersection trace lines, 
as shown by O 2 W M 3 and E P 2 . 
From G as center, with radius G 
M 1 , describe the arc L 2 M*. Then 
1/ H' O 3 1ST' is the pattern of the 
upper section, and O 2 N 3 P s E 2 is 
the pattern of the third section. 

51S. Three-Piece Elbow m 
Flaring Pipe, the Middle Section 
of which is Straight,— -In Fig. 380, 
let F G B A D C H I be an eleva- 
tion of the elbow, the lower section 
FGHI and the upper section 
B A D C of which both flare, 
while the middle section B C H G is 
straight. For the patterns we pro- 
ceed as follows : Produce the sides 
Then E is the apex and E B and E C sides 




Fig. 



7 

380. — Three-Piece Elbow in Flaring Pipe, the Middle Section of which is Straight. 



B A and C D of the upper section until they meet in the point E. 
of a cone, of which A D C B is a section. At any point outside of the section, at right angles to the axis 
TJ E, draw L Iv, and produce the sides E B and E C until they meet it, as shown by E L and E K. In line 
with the middle section draw the profile M P Y, as shown. Divide MPTiu the usual manner into any con- 
venient number of equal parts, as shown by the small figures. Through the points thus obtained carry lines 
vertically, cutting the miter line B C. Construct a section of the cone as it would appear if cut on the line 
L K as follows : Produce the axis TJ E to any convenient point, as O'. From O 1 as center, with radius equal 
to M, as shown by O 1 M 1 , describe the semicircle M 1 P 1 Y 1 . Divide this semicircle into the same number of 
equal parts, as M P Y already described, and through the points from O 1 draw radial lines indefinitely, as shown 
by O 1 1, O 1 2, 0' 3, etc. Through the points in B C already obtained draw lines from the apex E, cutting L K, 
and thence, parallel to the axis, drop points intersecting the lines drawn from O 1 . Through the points thus 
obtained trace a line, as shown by L' S 2 K 1 , which will be the profile of the cone when cut on the line L K. 
From the points in B C also, at right angles to the axis U S, draw lines cutting F S, as shown by the small 
figures, 0, 1, 2, 3, etc. At any convenient point draw E' S 1 , Fig. 381, equal to E S of the elevation. Set off 
E 1 U l equal to E U of the elevation. Likewise set off points, as shown by the small figures, corresponding to 
the points in II S of the elevation. From S 1 , at right angles to E 1 S 1 , draw S 1 K 2 , equal to S K of the eleva- 



Pattern Problems. 



131 




0, 2 V 6 'K 3 
Fig. 381.— Pattern of Upper Section. 
Three-Piece Ellioro in Flaring Pipe, the Middle 
Section of which is Straight. 



tion. On this line, measuring from S 1 , set off distances corresponding to the length of the several radial lines 
between the center O 1 and the profile L' S = IC, all as shown by the small figures, 0, 1, 2, 3, etc. From these 
points draw fines to E 1 . At right angles to E 1 S 1 draw lines corresponding to II 1 and the points indicated by 
the small figures, which produce until they intersect the corresponding radial lines drawn from E 1 . From E as 
center, with radii corresponding to the several points in S 1 K 1 , describe arcs indefinitely. From E draw the 
straight line E 1 T, which will form the boundary of one side of the 
pattern. Commencing at 7, which is in the outer arc, step off the 
stretchout of the plan L' S" K 1 , stepping from 7 to the second are, 
as shown at 6, and from there to the third arc, as shown at 5, and 
so on, each time stepping to the next arc. From the points in the 
arcs thus obtained draw measuring lines to E 1 . From E 1 as center, 
with radii corresponding to the intersection between the lines drawn 
perpendicular to US and the radial lines drawn from E 1 to the 
points in S 1 K 1 , describe arcs intersecting measuring lines already 
drawn. Then a line traced through these points, as shown by C 1 B 1 , 
will form one boundary of the required pattern. From E 1 as center, 
with radius equal to E D of the elevation, or, what is the same, 
with radius corresponding to the intersection of the line drawn 
from FT, with a radial line corresponding to K 2 , describe the arc 
D 1 A'. Then D 1 A' P. 1 C 1 is half of the required pattern. For 
the pattern of the middle section, at right angles to its straight end, 
G H, lay off a stretchout taken from the plan M P T, as shown by 
IP G 2 , Fig. 380, through the points in which draw measuring lines in 
the usual manner. Place the T-square at right angles to this section of the pipe, or, what is the same, parallel to 
the stretchout line, and, bringing it successively against the points in B G, cut measuring lines of corresponding 
numbers. Then a line traced through the points thus obtained, as shown by B 2 C s , will be the shape of the pat- 
tern corresponding to the line B C in the elevation. For the lower section of the elbow produce the sides F G 
and I H until they meet in the point T. Then T is the apex, and T F and T I sides of a right cone, of which 

G II I F is a frustum. From the center O of the plan, which is in 
line with the axis of the cone, with radius equal to Z I of the elevation, 
describe the semicircle V "W" X, which will be the plan of the cone at 
the base. Divide V¥X into any number of equal parts for use in lay- 
ing off the stretchout. From any convenient center, as T 1 in Fig. 3S2, 
with radius equal to T F of the elevation, describe the arc F 1 I 1 indefi- 
nitely, upon which set off a stretchout of the plan Y "W" X in the usual 
manner. From T 1 as center, with radius equal to T G of the elevation, de- 
scribe the arc G 1 IP. From the last point in F' I 1 (13) draw a line toward 
the center T 1 , cutting the smaller arc in the point IF. Connect G 1 F'. 
Then G 1 F 1 I 1 II 1 is half the pattern of the lower section of the elbow. 

519. Three-Piece Plbow, the Middle Section of which Tajiers. — ■ 
In Fig. 383, let D E G L X M K F be an elbow, the middle sec- 
tion (F G L K) of which tapers, the upper and lower sections being 
straight. For the patterns proceed as follows : Opposite and in line 
with the upper straight section draw the half profile ABC, which 
divide in the usual manner into any convenient number of equal parts, 
as indicated by the small figures, 1, 2, 3, etc. Draw the miter line F G 
between the sections, and from the points in the profile ABC drop lines, 
cutting F G as shown. Opposite the end D E, and at right angles to 
that section of the pipe, lay off the stretchout D 1 E 1 of the plan ABC, through the points in which draw the 
usual measuring lines. "With the T-square placed parallel to this stretchout line, or, what is the same, at right 
angles to the lines of the section, and brought successively against the points in the miter line F G, cut corre- 
sponding measuring lines, as shown. Through the points thus obtained trace a line, as shown from F 1 to G'. 
Then D l E 1 G 1 F 1 will be the half of the required pattern of the upper section. For the pattern of the lower 




-'T 1 



Fig. 332. — Pattern of Lower Section. 

Three-Piece Elbow in Flaring Pipe, the 

Middle Section of which is Straight. 



132 



Pattern Problems. 




section proceed in the same general manner. Draw the half profile P O R in line with it, which divide 
into any convenient number of equal spaces, from the points in which carry lines vertically, cutting the miter 
line K L hounding the section. Opposite the straight end of this section, and at right angles to it, draw the 
stretchout line M 1 W, in length equal to the half section P O R. Through the points in M 1 N" 1 draw the usual 

measuring lines. Place the J-square 
at right angles to the lines of the sec- 
tion, and, bringing it successively 
against the points in K L, cut meas- 
uring lines of corresponding numbers, 
as shown. Then a line traced through 
these points, as shown by K 1 L 1 , will 
be the shape of the pattern of the 
lower pieces. For the pattern of the 
middle section proceed as follows : 
Produce the sides L G and K F until 
they meet in the point H. Then H 
is the apex, and H L and H K are 
sides of a cone of which F G L K is 
a section. Through the point Y, 
which represents the intersection of 
the axis of the upper section of the 
pipe with the miter line F G, draw 
H Y, which produce indefinitely in 
the direction of IT. At right angles to 
II IT, and at any convenient point out- 
side the section F6IK, draw T L. 
Produce the sides F K and G L until 
they meet this line. The next step is 
to construct a section of the cone as it would appear if cut on the line T L. Produce H IT, as shown by H S 1 . 
At right angles to II S 1 , through S 1 , draw the line T 1 L 1 indefinitely. From S 1 as center, with radius equal to 
S O of the plan of the lower section, describe the arc P 1 O 1 K, which 
divide into the same number of equal parts as the profile POP. 
From S 1 , through these points, draw radial lines indefinitely. Through 
the points in the miter line K L, obtained from the profile POE 
already described, draw lines from the apex H, cutting T L, and from 
this line carry them parallel to the axis II IT, until they intersect radial 
lines drawn from S 1 . Through these points of intersection trace a 
line, as shown by T' IT 2 L 1 . Then this line is the profile of the cone 
as it would appear if cut on the line T L. From the points in the 
miter line K L draw lines at right angles to the axis H IT, cutting 
H IT, as shown in the points 4 and 4 8. In like manner cut H IT by 
lines drawn at right angles to it from the points in F G, also shown by 
the points between 4 and 4 8. From any convenient point, as H 1 in 
Fig. 3S4, draw the line II 1 IP, in length equal to II IT of the eleva- 
tion. At right angles to IP F? set off IT' T', equal to IT T of the ele- 
vation. In H 1 U 1 set off points corresponding to the points in H IT in 
the elevation. With the dividers take the distance upon each of the 
several lines radiating from S 1 in the profile to the line T 1 FT 2 L 1 , and 
set off like distances from U 1 on FT 1 T", all as shown by the small figures from 8 to 0. From these points draw 
lines to H", intersecting them by lines drawn at right angles to H 1 U" 1 , from the points of like numbers in that 
line already described. Having thus obtained measurements of the middle section at the several points required 
they are spread, and the pattern itself is described as follows : From IP as center, with radius corresponding 
to the several points in FT 1 T 1 , describe arcs upon which to lay off the stretchout of the profile T 1 IT 9 L 1 . Draw 



Fig. 3S3. — Three-Piece Elbow, the Middle Section of which Tapers. 




Fig. 384.— Pattern of Middle Section. 

Three-Piece Elbow, the Middle Section of 

which Tapers. 



Pattern Problems. 



183 



any straight line from H 1 , as shown by H' 0. Set the dividers to the space 1 in the section T 1 U" L 1 , and 
commencing at in Fig. 384, step to the second arc, and from the point last set off step to the third arc 
and thus continue until the stretchout F 1 TJ 1 L 1 lias been laid off, stepping from arc to arc, as described. 
From the points in the stretchout thus obtained draw measuring lines to the center II 1 , all as shown. From H 1 
as center, with radii corresponding to the intersections of the lines drawn perpendicular to IT IT 1 with the radial 
lines drawn from H' to TJ 1 T', describe ares, which produce until they intersect measuring lines of correspond- 
ing numbers, all as indicated in the engraving. Then lines traced through the points of intersection thus 
obtained, as shown by F ! G 3 L 2 K°, will be the pattern sought. 

520. A Two-Piece Elbow in Elliptical Pipe. — The only difference to be observed in cutting the patterns 
for elbows in ellip>tical pipes, as compared with the 
same operations in connection with round pij>es, lies 
with the profile or section. The section is to be placed 
in the same position as shown in the rules for cutting 
elbows in round pipe, but it is to be turned broad or 
narrow side to the view, as the requirements of the 
case may be. In round pipe there is, of course, no 
such distinction possible. In Fig. 385 is shown a right 
angled two-piece elbow in an elliptical pipe, which 
shows the flat side to the front. The same rule would 
apply in cutting the patterns if the elbow occurred in 
a pipe showing the narrow side to the view, the only 
change being in the placing of the section. The dem- 
onstration which follows, together with the reference 
given above to the rules for cutting elbows in round 
pipe, will be sufficient to enable the mechanic to cut 
the patterns of any required elbow in elliptical pipe. 
Let ACEFDB be the elevation of the elbow at 
the required angle. Draw C D, which forms the 
miter line. In line with one arm of the elbow draw 
a section, as shown by G H I K, which divide in the 
usual manner, and by means of the T-square placed 
parallel to the arm, drop points upon the miter line, as 
shown. Opposite the end of the arm lay off a stretch- 
out, and through the points in it draw the usual 
measuring lines. Reversing the J-square, placing it 
at right angles to the arm, and bringing it in con- 
tact with the several points in the miter line, cut 
the corresponding measuring lines. A line traced Fig. 385.—^. Two-Piece Elbow in Elliptical Pipe. 
through these points, as shown by L F O, will constitute the required pattern. 

521. A "[-Joint between Pipes of the Same Diameters. — Let DFGHHIKEin Fig. 3S6 represent a junc- 
tion between two pipes of the same size at right angles, of which ABC and A 1 B' C are sections. As the two 
pipes have like sections, the miter line's F L and K L appear straight in elevation. Space both sections into the 
same number of equal parts, as shown, and drop points on to the miter lines. Lay off two stretchouts, IS!" at 
right angles to the upper pipe and R T at right angles to the lower pipe. Set the T-square at right angles to 
the upper pipe, and, bringing the blade against the several points on the miter lines, cut the corresponding 
measuring lines drawn through the stretchout, as indicated by the dotted lines. Then N F 1 IT Y W will be 
the pattern for the upper piece. By inspection of the elevation and sections it will be seen that only a portion 
of the measuring lines are required to be drawn through the stretchout R T. It will be noticed that 7 comes 
at the middle of the required opening, while 4 represents the position of the edges. Therefore, draw the lines 
4, 5, 6, 7, 6, 5, 4, as shown. Place the blade of the J-square at right angles to the lower section of pipe, and, 
bringing it against the several points in the miter lines, cut the corresponding measuring lines, as shown by the 
dotted lines. A line, X Y Q 0, traced through these points will bound the opening to be cut in the pattern 
for the lower pipe. For the pattern of the pipe, from the points 1 in the stretchout draw the lines R P and 




134 



Pattern Problems. 



T S, in length equal to the length of the pipe. Connect P S. Then PETS will he the required pattern. 
The seam in the pipe may he located as shown in the engraving, or at some other point, at pleasure. 

522. A J-Joint between Pijxs of Different Diameters. — In Fig. 387 it is required to make a joint at right 
angles between the smaller pipe D F G E and the larger pipe HKLI. For this purpose both a side elevation 
and an end view are necessary. At a convenient distance from the end of the smaller pipe in each view draw 
a section of it. Space these sections into any suitable number of equal parts, commencing at corresponding 
points in each, and setting off the same number of spaces, all as shown by A B C and A' B 1 C. From the 
points in A B C draw lines downward through the body of the large pipe indefinitely. From the points in 



B 4 




A 1 B 1 C 1 drop points on to the profile of 
the large pipe, as shown by the dotted 
lines. For the pattern of the smaller pipe 
take the stretchout of ABC, or, what is 
the same, A' B 1 C, and lay it off at right 
angles opposite tbe end of the pipe, as 
shown by "V "W. Draw tbe measuring lines, 
as shown. Then with the T-square set par- 
allel to the stretchout line, and brought suc- 
cessively against the points between F 1 and 
G 1 upon the profile of the large pipe, cut 
corresponding measuring lines, as shown. 
Then a line traced through these points, as 
shown from X to T, will form the end of 
the pattern. For the pattern of the larger 

Fig. 3 &6.-A J-Joint between Pipes of the Same Diameter. p j pe the stretchout IS taken from the profile 

view F 1 G' L\ and laid off at right angles to the pipe opposite one end, as shown by N P. A corresponding 
line, M O, is drawn opposite the other end, and tbe connecting lines M 1ST and P are drawn, thus completing 
the boundary of the pattern. For the shape of the opening to be cut in the pattern, in spacing the profile of 
the large pipe F 1 G L 1 , the points 1, 2, 3 and 4 are made to correspond to the points dropped from the section 
of the small pipe, the other divisions of the profile being taken at will simply for the purpose of obtaining a 
correct stretchout. From these points (12 3 4) in the stretchout, therefore, measuring lines are drawn, inter- 
secting those previously dropped from corresponding points in the profile ABC, giving points through which 
the line E S T U is traced, which forms the shape of the opening. If for any reason it be desired to show a 
correct elevation of the junction between the two pipes, the miter line F G is obtained by intersecting the 
lines dropped from ABC with lines of corresponding numbers from F 1 G' in the profile of the large pipe. 



Pattern Problems. 



135 



523. A "[-Joint between Pipes of Different Diameters, the Smaller Pipe Setting to One Side of the Larger. 
— In Fig. 388, let A B C be the size of the small pipe and F 1 II' M 1 be the size of the large pipe, between 
which a right-angled joint is to be made, the smaller pipe being set to one side of the axis of the large pipe, as 
indicated in the profile. Draw an elevation, as shown by D F I L M K G E. Also draw a section, as shown 
by D 1 F 1 M 1 H 1 E'. Place a profile of the small pipe above each, as shown by ABC and A' B 1 C, both of 
which divide into the same number of equal parts, commencing at the same point in each. Placing the T-square 
parallel to the small pipe, and, bringing it successively against the points in the profile A' B 1 C 1 , drop lines cut- 
ting the profile of the large pipe, as shown from F' to H 1 ; and in like manner drop lines from the points in 
the profile ABC, continuing them 
through the elevation of the larger 
pipe indefinitely. For the pattern 
of the small pipe set off a stretch- 
out line, V W, at right angles to 
and opposite the end of the pipe, 
and draw the measuring lines, as 
shown. These measuring lines are 
to be numbered to correspond to 
the spaces in the profile, but the 
place of beginning determines the 
position of the seam in the pipe. 
In the illustration given we have 
located the seam at the shortest 
part of the pipe, or, in other words, 
at the line corresponding to the 
point 10 in the section. There- 
fore we commence numbering the 
stretchout lines with 10. Place 
the T-square at right angles to the 
small pipe, and, bringing the blade 
successively against the points in 
the profile of the large pipe from 
F 1 to IP, cut the corresponding 
measuring lines, as shown. A line 
traced through the points thus ob- 
tained, as shown by X Y Z, will 
form the end of the required pat- 
tern. For the pattern of the large 
pipe lay off a stretchout of the end 
view, locating the seam where de- 
sired, as above described in connec- 
tion with the small pipe. In this 
instance we have located the seam 
on a line corresponding to point 13 in the profile, 
commence with this number. 




Fig. 387.^.4 J-Joint. between Pipes of Different Diameters. 



Therefore, in laying off the stretchout, as shown on E, we 
Aiter laying off the stretchout opposite one end of the pipe, draw a correspond- 
ing line opposite the other, as shown by 1ST P, and connect 1ST O and P K, thus completing the outline. In spac- 
ing the profile of the large pipe, the spaces in that portion against which the small pipe fits are made to corre- 
spond to the points obtained by dropping lines from the profile of the small pipe upon it, as shown by 1 to 7 
inclusive. This is done in order to furnish points in the stretchout corresponding to the lines dropped from the 
profile A B C, as shown. Xo other measuring lines than those which represent the portion of the pipe which 
the small pipe fits against, are required in the stretchout. Accordingly the lines 1 to 7 inclusive are drawn from 
O E, as shown, and are cut by corresponding lines dropped from A B C. A line traced through the several 
points of intersection gives the shape STU, which is the opening in the large pipe. If it be necessary for 
any purpose to show a correct elevation of the junction between two pipes, the miter line FHGis obtained 



136 



Pattern Problems. 



by intersecting the lines dropped from ABC by corresponding lines carried across from the same points 
obtained on the profile F' H 1 , by dropping from A B C, as previously explained and all as shown by the dotted 
lines. 



521. A Joint between Two Pipes of the Same Plametei 




at Other than Righ t Angles.— Let LFDEKIHM 

of Fig. 389 represent the 
pipes 
angle 



elevation of two 
meeting in the 
MHI, for which pat- 
terns are required. Draw 
the profile or section 
A 1 B 1 C in line with the 
branch pipe, and the sec- 
tion A B C in line with 
the main pipe. Space 
both the profiles into the 
same number of equal 
divisions, commencing at 
the same point in each. 
Draw lines from these 
points, which produce 
until corresponding lines 
from the two sections in- 
tersect, and through the 
several points of inter- 
section thus obtained 
draw the lines F G and 
G H, which is the miter 
line between the two pipes. For the pattern of the arm pro- 
ceed as follows : Lay off the stretchout O X ojjposite the end 
of the arm, aud draw the usual measuring lines at right angles 
through it, as shown. Place the T-square at right angles with 
the arm, or, what is the same, parallel with the stretchout line, 
and, bringing the blade successively against the points in the 
miter line F G H, cut the corresponding measuring lines. 
Through the points thus obtained trace the line PEST, 
which will form the pattern required. For the pattern of the 
main pipe proceed as follows : Opposite one end lay off the 
stretchout, as shown by V T, and opposite the other end lay 
off a corresponding line, as shown by TF X. Connect ~U Y and 
X Y. From so many of the points in the stretchout line V Y 
as correspond to points in the miter line F G H, draw the usual 
measuring lines. Place the T-square at right angles to them, 
and draw lines from the points in the miter line F G H, inter- 
secting the corresponding measuring lines. A line traced 
through these points of intersection, as F 1 Z H 1 "W, will 
describe the shape required. The position of the seam in both the arm and the main pipe is determined 
by the manner of numbering the spaces in the stretchout. In the illustration the seam in the arm is 
located in the shortest part, or at a point corresponding to 1 of the profile. Accordingly, in numbering 
the divisions of the stretchout, that number is placed first. In like manner the seam in the main pipe 
is located at a point opposite the arm. Therefore, in numbering the spaces in the stretchout we commence 
at 1, which, as will be seen by the profile, represents the part named. If it were desirable to make the 
seam come on the opposite side of the main pipe from where we have located it — that is, come directly 
through the opening made to receive the arm — we would commence numbering the stretchout with 7. 



Fig. 3S8. — A J-Joint between Pipes of Different 
Diameters, the Smaller Pipe Setting to One 
Side of the Larger. 



Pattern Problems. 



137 



In that case the opening FfH'Z -would appear in two halves, and the shape of the pattern would be 
as though the present pattern -were cut in two on the line 7 and the two pieces were joined together on 1. 
By this explanation it will be seen that the seams may be located during the operation of describing the 
pattern wherever desired. 

525. The Joint between Two Pij>es of Different Diameters Intersecting at Other than Bight Angles. — Let 
ABC, Fig. 390, be the size of the 
smaller pipe, and TH'Z the size of 
the larger pipe, and let HLM be 
the angle at which they are to meet. 
Draw an elevation of the pipes, as 
shown by G K I O ST M L H, plac- 
ing the profile of the smaller pipe 
above and in line with the arm, as 
shown. Place an end view of the 
larger pipe in line with that part of 
the elevation, as shown, and directly 
above it, their center lines carre- 
sponding. Place a second profile of 
the small pipe, as shown by A 1 B 1 C. 
Divide both sections of the small 
pipe into the same number of spaces, 
commencing at the same point in 
each. From these points drop lines 
on to the large pipe, as shown, both 
in section and elevation. From the 
points thus obtained upon the pro- 
file of the large pipe carry lines 
across to the left, producing them 
until they intersect corresponding 
lines in the elevation. A line traced 
through these several points of in- 
tersection gives the miter line EL, 
from which the points in the two 
patterns are to be obtained. For 
the pattern of the small pipe pro- 
ceed as follows : Opposite the end 
lay off a stretchout, at right angles 
to it, as shown by E F. Through 
the points in it draw the usual 
measuring lines, as shown. Bring 
the T-square to right angles with 
the pipe, and, placing it successively 
against the points in the miter line 
K L, cut the corresponding measur- 
ing lines, as shown by the dotted 
lines. A line traced through the points thus obtained will give the pattern, as indicated. For the pattern of 
the large pipe proceed as follows : Opposite one end, and at right angles to it, lay off a stretchout, as shown 
by B S. Draw a corresponding line P T opposite the other end, and connect P B and T S. In order to afford 
corresponding points for measurement in describing the shape of the opening to be cut in the pattern of the large 
pipe, in spacing the profile, as shown by T N 1 Z, the points 4 3 2 12 3 4 are taken, as already established by 
the lines dropped from the profile of the small pipe. The other points in the profile are taken at convenience, 
simply for stretchout purposes. In laying off the stretchout B S that number is placed first which represents 
the point at which it is desired the seam shall come. For the shape of the opening in the pattern, draw measur- 




ing. 3Sg. — A Joint between 



Vu-o Pipes of the Same Diameter at Other than 
Bight Angles. 



138 



Pattern Problems. 



6 Z 



rug lines from the points i 3 2 1 2 3 4, as shown, and intersect them by corresponding lines dropped from the 
miter line. Through the points thus obtained trace the line U V W X, which will represent the shape of the 
opening required. 

526. A Joint at other than Bight Angles between Two Pipes of Different Diameters, the Axis of the 
Smaller Pipe being Placed to One Side of that of the Larger One. — In Fig. 391, let C B 1 A 1 be the size 
of the smaller pipe, and I)' E' I' the size of the larger pipe, between which a joint is required at an angle 
represented by W F K, the smaller pipe to be placed to the side of the larger. Draw an elevation of the pipes, 
joined as shown by Y D G II I K F "W". Place a profile or section of the arm in line with it, as shown by 

C 1 B 1 A 1 . Opposite and in line with the end of the 
main pipe draw a section of it, as shown by D 1 E 1 1 1 . 
Directly above this section draw a second profile of 
the small pipe, as shown by A B C, placing the cen- 
ter of it — relative to the center of the profile of the 
large pipe — in the same position that the arm is to 
have in the main pipe. Divide the two profiles of 
the small pipe into the same number of equal spaces, 
commencing at the same point in each. From the 
divisions in C 1 B 1 A 1 drop lines parallel to the lines 
of the arm indefinitely. From the divisions in 
ABC drop lines until they cut the profile of the 
large pipe, as shown by the points in the arc D 1 E 1 . 
From these points carry lines to the left, producing 
them until they intersect the corresponding lines 
from C B 1 A 1 . A line traced through these points 
of intersection, as shown by D E F, will be the miter 
line between the two pipes. For the pattern of the 
arm proceed as follows: Lay off a stretchout at 
right angles to and opposite the end of the arm, as 
shown by Ft P, and through the points in it draw 
the usual measuring lines. Place the T-square at 
right angles to the arm, and, bringing it successively 
against the points in the miter line, cut the corre- 
sponding measuring lines. A line traced through 
these points, as shown by TJ T S, will form the 
required pattern. For the pattern of the main pipe 
draw a stretchout line opposite one end of it, as 
shown by M O, numbering the divisions in it with 
reference to locating the seam, which can be jjlaced 
at any point desired. Draw a line corresponding to 
the stretchout line opposite the other end of the 
pipe, as shown by L N, and connect L M and 1ST O. 
Through as many of the points in the stretchout line as correspond to the points forming the miter line D E F 
in the elevation draw measuring lines, as shown by 1, 2, 3, 4, 5, 6 and 1. Place the T-square at right angles to 
the main pipe, and, bringing the blade against the points in D E F successively, cut the corresponding measur- 
ing lines, all as shown by the dotted lines. A line traced through these points of intersection, as shown by 
E 2 I) 2 F 2 , will give the shape of the opening to be cut in the pattern. 

527. The Patterns of a Cylinder (or Pipe) and Cone Meeting at Right Angles to their Axes. — In Fig. 
392, let B G E D F A C be the elevation of the required article. Draw the plan in line with the elevation ) 
making like points correspond in the two views, as shown by M S T U P 1ST. Draw a section of the pipe in 
proper position in both elevation and plan, as shown by E M 1 D and N D 1 M respectively. Divide these sec- 
tions of the pipe into any convenient number of equal parts, commencing at the same point in each, as shown 
by the small figures. From the center of the section of the pipe, as shown in plan, draw a straight line to the 
center of the plan of the cone, as shown by D 1 P. From each of the points in the section of the pipe shown 




Fig. 390. — The Joint between Two Pipes of Different Diameters 
Intersecting at Other than Bight Angles. 



Pattern Problems. 



139 



in elevation carry lines parallel to the sides of the pipe, cutting the side of the cone, and for convenience extend 
them some distance into the figure — for example, until they meet the axis. From the several points of inter- 
section with the side of the cone, as shown hy a b c d e, drop lines parallel to the axis of the cone, on to the 
line D 1 R of the plan, giving the 
points a 1 V c 1 d l e\ and through 
each of these points, from R as 
center, describe an arc, as indi- 
cated in the engraving. From 
the points in the profile 1ST D 1 M 
of the pipe in the plan draw 
lines parallel to the sides of the 
pipe, producing them until they 
meet the arcs drawn through 
corresponding points, as dropped 
upon D 1 E from the elevation, 
giving the points indicated by 
l 1 , 2 1 , 3 1 , i 1 and 5'. From these 
points carry lines vertically to 
the elevation, producing them 
until they meet the lines drawn 
from points of corresponding 
numbers in the profile of the 
pipe to the axis of the cone, giv- 
ing the points T, 2 2 , 3", 4 3 and 
5 3 . A line traced through these 
points, as shown from G to F, 
will be the miter line in eleva- 
tion formed by the junction of 
the pipe and cone. To describe 
the patterns proceed as follows : 
Opposite the end of the pipe, as 
shown in elevation, and at right 
angles to it, lay off a stretchout, 
K Ii, through the points in which 
draw the usual measuring lines. 
Intersect these measuring lines 
by lines from corresjjonding 



points, 1 , 
miter line, 



2\ 



etc., in 



as produced in 



the 
the 




Fig. 



391. — A Joint at other than Right Angles between Two Pipes of Different Diameters, 
the Axis of the Smaller Pipe being Placed to One Side of that of the Larger One. 



elevation. A line traced through 
these points of intersection, as 
shown from L to I, will be the 
shape of the end pipe to fit 
against the side of the cone, and 
the entire pattern of the piece 
will be as shown by Ii I L K. 
From any convenient point, as A 1 , Fig. 398, draw A 1 B 1 , in length equal to A B of the elevation. Set off 
points e", d~, c\ V and a" in it corresponding to e, d, c, b and a of A B, Fig. 392. From A 1 as center, with 
radius A 1 B 1 , describe the arc B 1 Y, upon which lay off the stretchout of the plan of the cone, as indicate*"" 
by the small figure outside of the pattern. (But one-half of the pattern is shown in the engraving.) Fron. 
the same center A 1 describe arcs corresponding to the points e% d% c% ¥ and a\ From the center It of 
the plan draw lines to the circumference through the points 2 1 , 3 1 , 4=\ etc., giving the points in the circum- 
ference marked 2 3 , 3 3 , 4 3 , etc. Set off corresponding points in the are B 1 V, as shown by 3', 2 4 , 4 4 , 5*, etc. 



uo 



Pattern Problems. 



From, these points draw lines to the center A', intersecting the arcs drawn from a% V, <?, etc. A line traced 



through 



these points of intersection, as shown by F 1 O 1 G 1 P 1 , will be the shape of the opening to cut to 
correspond with the pipe. 

52S. The Patterns of a Frustum of a Cone Intersecting a Cylinder, their Axes being at Eight Angles. — 

Let SPET in Fig. 394 be the elevation of the cylinder, 
and a G K II b the elevation of the frustum. Draw the 
axis of the cylinder, as shown by A B, which prolong, as 
shown by C D, on which construct a profile of the cylin- 
der, as shown by C E D F. Produce the sides of the 
frustum, as shown in the elevation, until they meet in the 
point L, which is the apex of the cone. Draw the axis 
L K, which produce in the direction of 0, and at any 
convenient point in the same construct a profile of the 
cone M 1ST as it would appear if cut on the line a 5. 
In connection with the profile of the cylinder draw 





Fig. 393.— Half Pattern of Cone. 
The Patterns of a Cylinder (or Pipe) arid Cone 
Meeting at Right Angles to their Axes. 



a corresponding elevation of the cone, as shown by 
K' a 1 V K\ Produce the sides K 1 a 1 and K? o' until they 
intersect, thus obtaining the point L 1 , the apex corre- 
sponding to L of the elevation. Draw the axis L 1 E, as 
shown, which produce in the direction of N 1 , and upon 
it draw a second profile of the cone taken on the line 
a b, as shown by M' O 1 N\ Divide the profiles ION 
and M 1 0' N 1 into the same number of equal parts, com- 
mencing at corresponding points in each, as shown. With 
the T-square set parallel to the axis of the cone, and 
brought successively against the points in the profile, 
drop lines to the lines a b and a 1 b\ as shown. Place the 
T-square against the apex L 1 , and, bringing it successively 
ao-ainst the points in a 1 b\ cut the profile of the cylinder, as shown in E? E K\ In like manner place the 
T-square against the apex L, and draw lines indefinitely through the points in a b. Place the T-square parallel 
to the sides of the cylinder, and, bringing it against the points in the profile K' E K? just described, cut corre- 
sponding lines in the elevation, as shown by H K G. A line traced through these points of intersection, as 
shown by II K G, will form the miter line between the two pieces as it appears in elevation. Continue the 
lines drawn from K 1 E K° until they meet the side a G of the cone prolonged, as shown from G to Z. From 



Fig. 392. — The Patterns of a Cylinder (or Pij^e) and Cone 
Meeting at Right Angles to their Axes. 



Pattern Problems. 



141 



L as center, and with radius L a, describe the arc If a\ upon which lay off a stretchout of the profile MONof 
the cone. Through each of the points in this stretchout draw lines indefinitely, radiating from L, as shown. 
Number the points in the stretchout a' V corresponding to the numbers in the profile, commencing with the 
point occurring where it is desired to have the seam. Set the compasses, with L for center, to L Z as radius, 
and describe an are cutting the corresponding lines drawn through the stretchout, as shown by 1, 5 and 1. In 
like manner reduce the radius to the second point in G Z, and describe an arc cutting 2, 4, 4 and 2. Also bring 
the pencil to the third point and 
cut the lines corresponding to it 
in the same way. Then a line 
traced through the points thus 
obtained, as shown by IT K s G 1 , 
will be the pattern of the cone. 
At right angles to and opposite 
one end of the cylinder draw a 
stretchout, taken from the profile 
C E D F, as shown by X V. In 
laying off this stretchout let 
points in the portion of the pro- 
file represented by E? E X 2 cor- 
respond to the divisions obtained 
by dropping lines from the apex 
of the cone. The other points 
in the profile may be taken at 
will, being used only for stretch- 
out purposes. In laying off the 
stretchout commence at a point 
corresponding to the place at 
which the seam is desired to be 

in the finished work, in this case at 9. Through the 
points 1, 2, 3, 4 and 5, being those in the portion of the 
profile over which the cone sets, draw measuring lines, 
as shown. The usual measuring lines may be dispensed 
with through the other points. Place the T-square at 
right angles to the cylinder, and, bringing it succes- 
sively against the points in the miter line, as shown in 
the elevation, cut the corresponding measuring lines. 
Then a line traced through these points of intersec- 
tion, as shown by G 2 X 4 IT X 5 , will be the opening to 
be cut in the pattern of the cylinder. Draw U W 
opposite the other end of the cylinder, as shown in 
elevation, parallel and equal in length to X Y, and 
connect U X and W V, thus completing the pattern 
of the cylinder. 

529. The Frustum of a Cone Intersecting a Cyl- 
inder of Greater Diameter than Itself at. Other than 
Eight Angles.— Ii\ Fig. 395, E G H F represents an elevation of the cylinder, and MKLK an elevation of 
the frustum of a cone intersecting it. F'ZQ represents the profile of the cylinder, or, what is the same, the 
cylinder in plan. Having drawn the elevation and the plan under it, as shown in the engraving, for the pat- 
terns proceed as follows : At any convenient point on the axial line of the cone, as indicated by T O, construct 
the profile Y Y X W, which represents a section through the cone on the line M X. Divide the section 
Y Y X "W into any convenient number of equal spaces in the usual manner, as shown by the small figures, 1, 
2, 3, 4, etc. From each of the points thus established drop lines parallel with the axis of the cone, cutting the 
line M X. From the intersections in M X thus obtained drop points parallel with the side G H of the cylinder, 




W 
Fig. 



394- — The. Patterns of a Frustum of a Cone Intersecting 
a Cylinder, their Axes being at Right Angles. 



U2 



Pattern Problems. 



and continue them indefinitely, cutting the line F' O 1 , which is drawn through the center of the plan of the 
cylinder at right angles to the elevation, all as shown in the engraving. Make Y 1 "W equal to Y TV" of the first 
section constructed. In like manner measure distances from the center line Y X of the first section to the 
points 2, 3, 4, etc., and set off corresponding spaces in the second section, measuring from M 1 If', upon lines of 
corresponding numbers dropped from the intersections in M If already described. Then a line traced through 
these points will represent a view of the upper end of the frustum as it would appear when looked at from a 
point directly above it. Produce the sides of the frustum K II and L If until they meet in the point 0. From 
drop a line parallel to the side G II of the cylinder, cutting the line F 1 O 1 in the point O 1 , thus establishing 
the position of the apex of the cone in the plan. From the point O 1 thus established draw lines through the 
several points in the section M 1 Y 1 If 1 W, which produce until they intersect the plan of the cylinder in points 
between Z and Q, as shown in the engraving. From O, the apex of the cone in the elevation, draw Hues through 




Fig. 395. — The Frustum of a Cone Intersecting a Cylinder of Greater Diameter than Itself at Other than Bight Angles. 

the several points in M. If already determined, which produce until they cross G H, the side of the cylinder, 
and continue them inward indefinitely. Intersect these lines by lines drawn from the points between Z and Q 
of the plan just determined. Then a line traced through these intersections, as indicated by K T L, will rep. 
resent the miter between the frustum and cylinder as seen in elevation. "With this line determined we are now 
ready to lay off the patterns, to do which proceed as follows : From O as center, with If as radius, describe the 
arc P P, on which set off a stretchout of the section Y Y "W X in the usual manner. From O, through the 
several points in P P thus obtained, draw radial lines indefinitely. From the several points in the miter line 
K T L draw lines at right angles to the axis O T of the cone, producing them until they cut the side If L. 
From O as center, with radii corresponding to the several points in If L just obtained, describe arcs, which pro- 
duce until they intersect radial lines of corresponding number drawn through the stretchout P E. Then a line 
traced through these points of intersection, as indicated by S L 2 U, will be the lower hue of the pattern souo-lit, 
and P S L' U E will be the complete pattern. For the pattern of the cylinder and the opening in it proceed 
as follows : Draw the line B D at right angles to the cylinder and in line with one end of it, upon which set 
off a stretchout of the cylinder from the plan F 1 Z Q in the usual manner. The points between Z and Q of the 
plan, as indicated by Z" Q 2 of the pattern, must be made to correspond with the divisions in the plan. From 
these points lines are to be drawn perpendicular to the stretchout line B D. Then, with the T-square placed at 



Pattern Prdblemcs. 



143 



right angles to the cylinder, and brought successively against the points in the miter line KTL, cut lines of 
corresponding numbers. A line traced through the points of intersection thus formed, as shown by Z' B? Q 1 L 1 , 
will be the shape of the required opening in the cylinder. 

530. The Patterns of the Frustum of a Cone Joining a Cylinder of Greater Diameter than Itself at 
Other than Eight Angles, the Axis of the Frustum passing to One Side of the Axis of the Cylinder. — Let 
E F H G in Fig. 396 be the elevation of a cylinder, which is to be intersected by a cone or frustum of a cone, 
DAIC, at the angle FDA in elevation, and which is to be set to one side of the center, all as shown by 
S O P L M R of the plan. Opposite the end of the frustum, in both elevation and plan, construct a section of 
it, as shown by T U V W in the elevation and T 1 IT 1 V W in the plan. Divide both of these sections into 
the same number of equal parts, commencing at corre- 
sponding points, and number them as shown by the 
small figures in the diagram. From the points in 
T UV¥ carry lines parallel to the axis of the cone, 
cutting the line A I, and thence drop them vertically 
across the plan. From the points in the section 
tji y> "^> draw lines parallel to the axis of the cone, 
as seen in plan, intersecting the lines dropped from 
A I described. Through these points of intersection 
trace a line, as shown by L M. Then L II will show 




the end of the frustum A I as it appears in plan. 
Through the points in L M draw lines cutting the plan 
of the cylinder, as shown from P to P. From the 
points of intersection between these lines and P R of 
the plali of the cylinder carry lines vertically, inter- 
secting those in the elevation drawn from the apex X. 
Then a line traced through these points, as shown by 
K D C, will be the miter line in elevation. For the 
pattern of the cylinder lay off the stretchout G 1 LP 
in length equal to S O P P of the plan, in which set 
off points corresponding to the points used in the jtlan 
between P and R. The other lines appearing in G 1 LP 
are used merely for the purposes of a stretchout. 
Through these points, specially named above, and which in numbers are from 1 to 7 inclusive, draw lines at 
right angles to the stretchout line, as shown. "With the T-square placed at right angles to the axis of the cylin- 
der, and brought successively against the points in the miter line D K C, cut these lines in the manner indicated 
by the dotted lines. Then a line traced through these points of intersection, as indicated by D 1 K? C K 1 , will 
be the shape of the opening to be cut in the pattern of the cylinder to correspond with the intersection of the cone. 
Draw G' E 1 equal to G E of the elevation, and LP F' equal to II F of the elevation, and connect E' F', thus 
completing the patterns of the cylinder. For the pattern of the frustum, from any convenient center, as X, 
with radius X A, describe the arc A 1 I 1 , upon which lay off a stretchout of the section WTUT, through the 



Fig. 396. — The Patterns of the Frustum of a Cone Joining a Cylin- 
der of Greater Diameter than Itself at Other than Bight Angles, 
the Axis of the Frustum fiassing to One Side of the Axis of the 
Cylinder. 



144 



Pattern Problems. 



points in which, from X, draw radial lines indefinitely. Intersect these lines by arcs drawn from X, with radii 
corresponding to points in the side A I) produced, as shown from D to B. These points are obtained by lines 
drawn at right angles to the axis of the cone from the several points of intersection between the side D C and 
the lines drawn from X through the points in A I, all of which is clearly indicated by the dotted lines in the 

diagram. Through the points of intersection in the pattern thus obtained trace a line, 
as shown by C K 2 D'. Connect D 1 I 1 and C A 1 . Then D 1 C A 1 I 1 will be the pat- 
tern of the frustum D A I C, mitering with the cylinder at the angle described. 

531. Patterns of a Cylinder Joining a Cone of Greater Diameter than Itself at 
Other than Eight Angles. — Let B A K in Fig. 397 be the elevation of a right cone, 
perpendicular to the side of which a cylinder, LSTM, is to be joined. The first ope- 
ration will be to describe the miter line as it would appear in elevation. Draw the 
plan IT V AY of the cylinder, which divide into any convenient number of equal parts, 
as indicated bv the small figures, and from these points drop lines, cutting the side A K 
of the cone in the points H, F and D, producing them until they cut the axis A X in 
the points G E and C. The next step is to construct sections of the cone as it would 
:ar if cut on the lines G II, E F and C D. Draw a second elevation of the cone, 

as shown by B 1 A 1 E?, representing the cone turned 
quarter way around ; or, the first may be regarded 
as a side elevation and this as an end elevation. 
Draw a plan under the side elevation of the cone, 
as shown by X B. P 0, which divide into any con- 
venient number of equal parts, and in like manner 
draw a corresponding plan under the end elevation, 
as shown by Ft 1 P l O 1 X 1 . Divide this second plan 
into the same number of equal parts, commencing to 
number them at the same point as in the other plan. 
From the points 1 to 4 in plan X B P 0, carry lines 
vertically to the base B K, and thence toward the 
apex A, cutting the lines C D, E F and G II. In 
like manner, from the same points (1 to 4 inclusive) 
in the plan B 1 P 1 O 1 X 1 , carry vertical lines to the 
base B 1 K 1 , and thence to the apex A 1 . Place the 
T-square at right angles to the axes of the two 
cones, and, bringing it against the points of inter- 
section of the lines from B K with C D, cut corre- 
sponding lines in the second elevation, and through 
the points of intersection thus established" trace a 
line, as shown by M 1 M\ Produce the axis X 1 A 1 
to any convenient distance, upon which set off C D 1 , 
in length equal to C D, in which set off the points 
corresponding to the points in C D, and througb 
these points draw lines at right angles to C 1 D 1 . 
Place the T-square parallel to the axis X 1 A 1 , and, 
bringing it against the several points in M 1 M 3 , cut 
the lines drawn through C D 1 , as shown, and through 




397. — Patterns of a Cylinder Joining a Cone of Greater 
Diameter than Itself at Other than Right Angles. 



the intersections thus established trace a line, as 



shown by M 3 D 1 M 4 . Then M a D 1 W is a section 
of the cone as it would appear if cut on the line D. In like manner carry lines from E F across to the sec- 
ond elevation and thence parallel to the axis, cutting lines drawn through E 1 F 1 , which with its points is equal 
to E F, by which to establish the profile M 5 F 1 M°, which is a section of the cone as it would appear if cut on 
the line E F. Also use the points in G H in like manner, establishing the profile W II 1 M 8 , which represents a 
section of the cone as it would appear if cut on the line G H. (The lines indicating the operation in connection 
with the sections corresponding to E F and G H are omitted in the engraving to avoid confusion ; the opera- 



Pattern Problems. 



145 



Then a line traced through these 



tion is identical with that explained in connection with C D.) Having thus obtained sections of the cone corre- 
sponding to the several lines C D, E F, G II, arrange them together in line with the side KA of the cone 
placing the points D 1 F' II 1 tangent, all as indicated by C D 2 , E 2 F a , G 2 LP. In connection with these sections 
draw a plan of the cylinder, as shown by 1/ S 2 T 2 M 2 , opposite the end of which draw a profile, as indicated by 
FT 1 V W\ which divide into the same number of equal parts as used in the divisions of the profile U V ~W, 
commencing the division at corresponding jioints in each. From the points in the profile U' V W drop lines 
against the several profiles C 2 D 2 , E 2 F 2 and G 2 II 2 , arranged together, and thence drop the points back on 
to the elevation, cutting corresponding lines in it. That is, from the intersection of the line drawn from j>oint 
4 in U 1 V W 1 with the profile C 2 D 2 cut the line C D, which in the elevation corresponds to the point 4 in the 
profile U V "W, and from the intersection of a line drawn from 3 with E 2 F 2 cut the line E F, and so on, all as 
indicated by the dotted lines. Then a line traced through these points of intersection, as shown by L M, will 
be the miter line in elevation, after which the patterns are readily obtained, as follows : For the rjattern of the 
cylinder lay off a stretchout of the profile U VffS 1 T', opposite the end S T, through the points in which 
draw the usual measuring lines. Place the T-square at right angles to the same, and, bringing it a°-ainst 
the points in the miter line L M, cut the corresponding measuring lines, 
points, as shown from L 1 to M 1 , will be the shape of the pattern of 
the cylinder to fit against the cone. For the pattern of the cone, from 
any convenient center, as A 2 in Fig. 39S, with radius A B, describe the 
arc B 2 K 2 , which in length make equal to the circumference of the plan 
NEPO. From the apex of the cone, through such points in the miter 
line L M as do not correspond with lines already drawn, draw lines cut- 
ting the base B K, and thence drop them on to the plan SEPO, all 
as indicated by a, b and c. Set off in the arc B 2 K 2 points correspond- 
ing, as indicated by a 1 b 1 c l and « 2 V c°. From these points draw lines to 
the center A 2 , as shown, and also likewise from other points correspond- 
ing to points obtained in the miter line in the elevation. From A 2 as 
center, with radii corresponding to the points L, H, F, D and M of the 
elevation, strike arcs intersecting the lines just drawn, all as shown by 
1/ L 5 , IP H 6 , F 4 F s , etc. Then a line traced through the intersections 
thus obtained will be the shape of the opening to be cut in the envelope f ; s- 398. -Envelope of cone. 

of the COne corresponding to the Cylinder. The Patterns of a Cylinder Joining a Cone 

532. The Patterns of Two Cones of Unequal Diameter Intersecting °f ^^ Diameter than itself at other 
at Right Angles to their Axes. — Let U T T in Fig. 399 be the elevation 

of a cone, at light angles to the axis of which another cone or frustum of a cone, F G P, is to miter. Let 
LKNMbea section of the frustum on the line F G. Let IP W V 2 W 1 be a plan of the larger cone at the 
base. The first step in describing the patterns is to obtain the miter line in the elevation, as shown by the 
curved line from to P. "With this obtained the development of the pattern is a comparatively simple opera- 
tion. To obtain the miter line O P we proceed as follows : Divide the profile LlvNll into any convenient 
number of equal parts, as shown by the small figures. Inasmuch as the divisions of this profile are used in the 
construction of the sections — or, in other words, since sections must be constructed to correspond to certain lines 
through this profile — it is desirable that each half be divided into the same number of equal parts, as shown in 
the diagrams. Thus 2 and 2, 3 and 3, 4 and 4 of the opposite sides correspond, and sections, as will be seen in 
the upper part of the diagram, are made to agree with them. From the points thus obtained in the profile 
draw lines cutting the end F G of the frustum. Produce the sides O F and P G until they meet in E, which 
is the apex of the cone. From the points in F G draw lines from E, producing them until they cut the axis of 
the cone, as shown by A A 1 A 2 . Next construct sections of the cone as it would appear if cut through upon 
lines corresponding to these points, as A C, A' B, A 2 D. Divide the plan U 2 W V 2 W into any convenient 
number of parts. From the points thus established carry lines vertically to the base line U Y, and thence 
carry them to the apex T, cutting the lines A C, A 1 B, A 2 D, all as shown. Through each of the several points 
of intersection in these lines draw horizontal lines from the axis of the cone to the side, all as shown. At right 
angles to the lines A C, A 1 B, A 2 D draw lines to any convenient point, at which to construct the required sec- 
tions. Upon the lines drawn from the points A, A 1 , A 2 , at convenience, locate the points A 3 , A 4 , A 6 . Inasmuch 
as A 1 B is at right angles to the axis of the cone, the section corresponding to it will be a semicircle whose 




146 



Pattern Problems. 



radius will be equal to A' B. Therefore, from A 3 as center, with radius A' B, describe the semicircle S B 1 R. 
For the section corresponding to A* D lay off from A 6 the distances A 6 S 2 and A 6 B 2 , in a line drawn at right 




Fig. 399. — The Patterns of Two Cones of Unequal Diameters 
Intersecting at Right Angles to their Axes. 



to A" D of the elevation, each in length 
equal to the horizontal line drawn through the 
points in A 2 D from the axis to the side of the 
cone. At right angles to S 2 B 2 draw A 5 D 1 , in 
length equal to A 2 D of the elevation. Set off 
in it points 5 and 3, corresponding to similar 
points in A 2 D of the elevation. Through these 
points 5 and 3, at right angles to A 6 D 1 , draw lines 
indefinitely. From A 5 as center, with radius equal 
to the length of horizontal line passed through 
point 5, A 2 D of the elevation, describe an arc 
cutting line 5 drawn through A 5 D 1 . From the 
same center, with a radius equal to the length of 
the horizontal line drawn through point i in the 
line A 2 D of the elevation, strike an arc cutting 
the line 3. Then 



a line traced through these 



points, as shown by S 2 D 1 B 2 , will be the section of the cone as it would appear if cut on the line A 2 D of the 
elevation. In like manner obtain the section S 1 C 1 B 1 , corresponding to A C of the elevation. Brolong A 6 D 1 , 



Pattern Problems. 



117 



as shown by E 3 , making A 5 E 3 in length equal to A 1 E of the elevation. In like manner make A 4 E 2 and A 3 E 1 
equal to A E and A' E of the elevation respectively. At right angles to these lines in the sections set off F 2 G 2 , 
F 3 G 3 , F 4 G 4 , in position corresponding to F G of the elevation. Make the length of F 3 G 3 equal to the length 
across the section of the frustum marked 2 2. In like manner make F 3 G 3 equal to 3 3, and F' G 4 equal to 4 4 
of the section. From E 1 , E 3 and E 3 respectively, through these points in the several sections, draw lines. From 
the several points of intersection between the lines drawn from E 1 , E 2 , E 3 of the sections of the cone, as shown 




Fig. 400. — The Patterns of Two Frustums of Cones of Unequal Diameters Intersecting at Other than Eight Angles to their Axes. 

by d d, c,bb, carry lines back to the elevation, intersecting the lines A C, A 1 B, A 3 D. Then the line traced 
through these several intersections, as shown from O to P, will be the miter line in elevation. Having thus 
obtained the miter line, we proceed to describe the patterns as follows : For the envelope of the small cone, from 
any convenient center, as E, with radius E F, describe the arc F 1 G 1 , upon which set off the stretchout of the 
section KMN1. Through the points in this arc, from E, draw radial lines indefinitely. From E as center, 
with radii corresponding to the several points in the miter line P, cut the corresponding radial lines, as indi- 



148 



Pattern Problems. 



eated by the dotted lines. Then a line traced through these points of intersection, as shown by P 1 O 1 P 2 , will 
l)e the shape of the pattern to fit against the larger cone. For the pattern of the larger cone, from any con- 
venient point, as T, as center, with radius T IT, describe the arc V XT' in length equal to the circumference of 
the plan U¥?W ! of the cone. Upon this arc, V U 1 , set off points corresponding to the points from which 
lines were drawn to the base and thence to the apex. For obtaining measurements in connection with the miter 
line, through these points, 13 2 12 3 1, draw lines to the apex T. Intersect these lines by arcs struck from T 
as center, with radii corresponding to the points in the side of the cone between O 1 and P 3 , corresponding to 
the intersections of the lines drawn from the section L K M 1ST of the frustum. Then a line traced through 
these intersections, as shown by X Y Z, will be the shape of the opening to be cut in the envelope of the 
larger cone, over which the smaller cone will fit. 

533. The Patterns of Two Frustums of Cones of Unequal Diameters Intersecting at Other than Right 

Angles to their Axes.— In Fig. 400, let M N P O be the 
side elevation of the larger frustum, and F 1 G 1 S E the side 
elevation of the smaller, the two joining upon some line to 
be drawn from R to S. Produce the sides S G 1 and R, F 1 
until they meet in the point E. At any convenient place 
in line of the axis of the smaller frustum draw the profile 
II F K G, corresponding to the end F 1 G\ Divide this 
profile into any convenient number of ecpial parts, as shown 

by the small figures, 1, 2, 3, 
etc., and from these divis- 
ions, parallel to the axis of 
the cone, drop points on 
to F 1 G\ From the apex 
E, through these points in 
F 1 G 1 , carry lines, cutting 
the side ]ST P of the larger 
frustum, and producing 
them until they meet the 
opposite side, or, as in this 
case, the base O P, all as 
shown by B A, C A' and 
D A'. The next step is to construct sections of the 
larger frustum as it would appear if cut on each of 
these lines, from which to obtain points of measure- 
ment for determining the miter line from R to S in 
the elevation. Draw the plan of the base of the 
larger frustum, as shown by T II Y ¥, and divide 
one-half of it in the usual manner. From these 
jwints carry lines vertically to the base O P of the 
frustum. Produce the sides O M and P N until 
they meet in the point L. From the points in the 
base obtained from the plan carry lines to the apex 
L, cutting the section line A B, A 1 C and A ! D, as 
shown. Parallel to A B and of the same length, at 
any convenient point outside of the elevation, draw 
A 3 B 1 , and from the points in A B, obtained by intersections with the lines from the base O P to the apex L, 
draw lines at right angles, cutting it as shown in the points 7, 6, 5, 4, etc. In like manner make A 4 C 1 equal 
and parallel to A C, and from the points in A C draw lines at right angles to it, cutting it as shown, giving the 
points 5, 4, 3, etc. Also make A 6 D 1 equal to the section line A 2 D of the elevation, and by drawing lines from 
the points in it cut A 6 D 1 in the points 3, 2, 1, etc., as shown. In order to complete these several sections, the 
width of the frustum through each of the points indicated is to be set off on corresponding lines drawn through 
C 1 and A 6 D 1 . To obtain the width through these points chaw an end elevation of the article, as 




Fig. 4oi.— A Portion of the Preceding 
Figure brought forward, Showing the 
Method of Developing the Pattern of 
the Larger Frustum. 



The Patterns of Two Frustums of Cones of Unequal 
Diameters Intersecting at Other than Right Angles 
to their Axes. 



A B 1 , A 



Pattern Problems. 



149 



shown by M 1 1ST 1 P 1 O 1 . Produce the sides, obtaining the apex L 1 . Draw a plan and divide it into the same 
number of spaces as that shown in T IT Y W, and commence numbering at a corresponding point, all as indi- 
cated by V U 1 T" W. From the points in the plan carry lines vertically to the base O 1 P l , and thence to 
the apex L 1 . Place the blade of the T-square at right angles to the axis of the cone, and, bringing it succes- 
sively against the points in the section line A B in the side elevation, draw lines cutting the axis of the end ele- 
vation, and cutting the lines corresponding in number to the several points in A B, all as shown by a a, bb,cc, 
etc. Make the length of the lines drawn through A 3 B 1 equal to the corresponding lines thus obtained, as 
shown by a 1 a 1 , IS If, & c\ a" d\ etc., and through these extremities trace a line, as shown by/" 1 B'y, which will 
be the section through the cone when cut on the line A B. In like manner obtain l l C l l and/"' ~D l f". Pro- 
duce A 3 B 1 , making B 1 E 1 equal to B E of the elevation, and B 1 X 3 equal to B X 2 of the elevation. In like 
manner make C E 2 equal to C E, and C 1 X 4 equal to C X. Make D' E 3 equal to D E, and D 1 X 5 equal to D X. 
Through X 3 , at right angles to B' E 1 , draw a line in length equal to the line 2 2 drawn across the jirofile F K G IT, 
with which this section corresponds, as shown by 2 1 2 1 . In like manner, through X 4 draw a line equal to II K, 
as shown by H 1 K 1 , and through X 3 draw 4' 4 1 , in length equal to the line 4 4 drawn through the profile FK6E 
From E 1 , through the extremities of 2 1 2 1 , draw lines cutting the section. In like manner draw lines from E 2 
through the points II 1 E7, and from E 3 , 
through the points 4 1 4 1 . From the points 
at which these lines meet the profiles of the 
sections, a 1 a 1 in the first, o o in the second, 
and m 1 m' in the third, carry lines at right 
angles to and cutting the corresponding sec- 
tion lines in the elevation. A line traced 
through the points thus obtained, as shown 
by B S, is the miter line in elevation formed 
by the junction of the two frustums. Hav- 
ing thus obtained the miter line in eleva- 
tion, we proceed to develop the patterns as 
follows : From the points in B S, at right 
angles to A 1 E, which is the axis of the 
smaller cone, draw lines cutting the side 
E S, as shown by the small figures, 1. 2, 3, 
4 and 5. These points are to be used in 
laying off the pattern of the smaller frus- 
tum. From any convenient point for cen- 
ter, as E, with radius E G 1 , describe the arc F 2 G 2 , upon which stej) off the stretchout of the profile F K IT G, 
numbering the points in the usual manner. Through the points, from the center E, draw radial lines indefi- 
nitely. From the same center, E, with radius E 1 (of the points in E S), cut the radial line numbered 1, and 
in like manner, with radii E\ E 3 , etc., cut the corresponding numbers of the radial lines. A line, B 1 S l , traced 
through the several points of intersection thus formed will be the larger end of the pattern for the small frus- 
tum, thus completing the shape of that piece, all as shown by B 1 S 1 G 2 F 2 . To avoid confusion of lines, the 
manner of obtaining the envelope of the large frustum is shown in Fig. 401, which is a duplicate of the side 
elevation and plan shown in Fig. 400. The miter line B l S' and the points in it are obtained by transfer, being 
the same in all particulars as employed in the operations already described. Similar letters refer to correspond- 
ing parts in the several figures. From any convenient point, as L 2 , with radius L 2 O a , describe an arc, as shown 
by Y Z, and from the same center, with radius L 2 M 2 , describe a second arc, as shown by y s. Draw Y y, and 
upon Y Z lay off the stretchout of the plan TJ 2 Y' 2 "W" T 2 , all as shown. Draw Z z. Then Z z y Y will be the 
envelope of the large frustum. Through the points in the miter line B' S 1 draw lines from the apex of the 
cone to the base, and from the base continue them at right angles to it until they meet the circumference of the 
plan. Mark corresponding points in the stretchout Y Z, and insert any points which do riot correspond with 
points already fixed therein. From each of the points thus designated draw a line across the envelope already 
described to the apex, as shown by 6 IA 7 L 2 , 8 L 2 , 9 L 2 , etc. Also, from the points in the miter line B 1 S 1 
draw lines at right angles to the axis of the frustum, cutting the side L 2 O 2 , as shown. From L 2 as center, 
describe arcs corresponding to each of these points, and cutting the radial lines drawn across the envelope of 




Fig. 402. — Pattern for a Blower for a Grate, 



150 



Pattern Problems. 



the cone. A line traced through the points of intersection between arcs and lines of the same number, as 
shown by If H" 7r S 2 , will be the shape of the opening to fit the base of the smaller frustum. 

534. Pattern for a Blower for a Grate. — The blower shown in Fig. 402 consists of two pieces, the body 
and the hood, A section through the body, taken horizontally, shows an arc of an ellipse — a shape somewhat 
more flattened than a segment of a circle. The profile, taken through the blower vertically, shows the body 
straight, with the hood pitching toward the grate. L M O P is a profile through the blower, taken vertically at 
its center. A B C is a profile taken horizontally through the line F G II. DFIv II E is the elevation. Be- 
fore it is possible to cut the miters at the top and bottom of the piece F II Iv, a true stay of these pieces must 
be obtained, which is shown in connection with the side elevation. To obtain this stay proceed as follows : 
Divide one-half of the arc F K II into any number of equal spaces. Carry lines from each of these several 
points to the vertical line L X of the profile, and thence parallel with the line L X indefinitely. Intersect 
them at right angles by the line T S, located at any convenient point outside of the diagram. With the dividers 
take the horizontal distance between the points in the arc F K to the line K G, and set them off on the lines 

of corresponding number, measur- 
ing from the line T S. Then a line 
drawn through the points thus ob- 
tained, and as indicated by T B, will 
be a horizontal section through the 
correct profile of the inclined por- 
tion of the blower. Take the 
stretchout of the profile T B point 
by point, and place the spaces on the 
line U V, which is drawn at right 
angles to L M. Through the points 
in IT V draw the usual measuring 1 
lines at right angles to it. Drop the 
points from the profile T B on to 
both the miter lines M X and X L, 
and thence carry them, at right 
angles to L M, on to the stretch- 
out lines of corresponding numbers 
drawn from U Y. Then a line 
traced through the points thus ob- 
tained, and as indicated by F' E? H 1 , 
will be the desired pattern. 

535. Pattern of the Flaring End of an Oblong Pan. First Case — When both Bottom and Top 
of the Flaring End are Curved. — In Fig. 403, A B D C shows in elevation, and X P O B in plan, 
a vessel of the description indicated. To obtain the patterns, after having correctly drawn the plan and 
elevation, proceed as follows : Divide half of the boundary line of the bottom into any number of equal 
spaces, commencing at O, all as shown by the small figures 1, 2, 3, etc., in the plan. From the points thus 
obtained carry lines vertically until they cut the top line of the elevation, as shown in the points between B and 
L ; also continue the lines downward until they meet the line T 0, all as shown. From the points between L 
and B thus obtained draw lines parallel to B D, producing them upward indefinitely, and continue them down- 
ward until they meet the bottom line of the elevation F D, as shown. At right angles to the lines thus drawn, 
and at any convenient distance from the elevation, draw G II. "With the dividers, from the line G II, set off on 
each of the lines drawn through it the distance from T 0, on the lines of corresponding number, to the line rep- 
resenting the plan of the end. In other words, make G K equal to T G of the plan. Set off spaces on the 
other lines corresponding to the distance on like lines in the plan. Through the points thus obtained trace a 
line, as shown by K II. Then GHK will be the half profile of the end of the vessel at right angles to the 
line D B. The stretchout of the pattern is to be taken from the profile thus constructed. At right angles to 
D II, and at any convenient distance from it, lay off IT V equal to twice the length of K II, and make the 
divisions in it correspond with the divisions in K II. From the points in the stretchout thus obtained draw 
lines at right angles to it indefinitely. "With the blade of the T-square set at right angles with D B, and brought 




Fig. 



PLAN R 

403. — Pattern of the Flaring End of an Oblong Pan. First Case — When both 
Bottom and Top of the Flaring End are Curved. 



Pattern Problems. 



151 




Fig. 404.— Side, 



405.— End. 





successively against the points in F D, cut lines of corresponding numbers drawn through the stretchout. 
Then a line traced through these points, as shown by Z T, will be the pattern of the bottom of the end 
piece. In like manner, with the T-square in the same position, bring the blade against the points in L B and 
cut corresponding lines drawn through the stretchout. "What may be called the corner pieces of the pattern 
are to be added to the portion already obtained as follows : "With the dividers take the distance F E of the ele- 
vation as radius, and from the point Z of the pattern as center describe an arc. In like manner take the distance 
E L of the elevation in the dividers, and from the 
last point in the upper edge of the pattern already 
obtained, being that of the line 6, describe a sec- 
ond arc, cutting the one first drawn in the point 
W. Connect W Z, and also draw a line from W to 
the point in the line already obtained. Add a 
corresponding corner piece to the other extremity. 
Then Z W X Y will be the pattern required. 

536. Pattern of the Flaring End of an 
Oblong Pan. Second Case — When Top is 
Curved and Bottom is Straight. — In Fig. 404, 
ACDE represents the side elevation of the 
article, FHKL in Fig. 405 the end elevation, 
and MNEP in Fig. 406 the plan or bottom. 
By inspection of these it will be seen that the ^ g _ 4 o6.— pian. 

shape of the end piece required is such that it Pattern of the Flaring End of an Oblong Pan. Secotid Case— 

may be resolved into three parts or Sections. The When Top is Curved and Bottom is Straight. 

middle one of these will be flat, or as represented upon the end elevation by G L K. The two side pieces are 
sections of the envelope of a cone. To obtain the patterns proceed as follows : Divide one-half of the end of 
the plan into any convenient number of equal spaces, all as shown by small figures 1, 2, 3, 4, etc., in X' R. From 
each of the points thus determined draw lines to the point X, all as shown in the engraving. From the meas- 
urements made possible by these lines we next proceed to construct the diagram shown in Fig. 407. Draw A B, 
in length equal to D B of Fig. 404. At right angles to it draw B C, which produce indefinitely. From B 
alono- B C set off spaces equal to the distance from X, Fig. 406, measured to the points in the boundary line of 
the plan. That is, make B 5 of Fig. 407 equal to X 5 of Fig. 406, and B 4 equal to X 4, and so on. With 
the measurements to be obtained from this diagram we lay off the patterns as follows : Draw A 1 D, in length 

equal to A C of the diagram. Set off points in A 1 D to rep- 
resent the length of the lines in the diagram drawn from A, 
or, in other words, make A 1 2 equal to A 2 of the diagram, 
and so on. From A 1 as center, with radius A 1 D, describe the 
arc D E indefinitely. In like manner, from the same center, 
with radius A 1 2, describe a corresponding arc, and proceed in 
this way with each of the other points lying in the line A 1 D. . 
From A 1 , and at any convenient angle, draw A' E, letting E 
fall in the arc D E, already mentioned. From E, stepping 
from one arc to another, lay off the stretchout of X B of Fig. 
406, all as shown by E F of the pattern. Connect A F. Then 
A 1 F E will be one section of the required pattern. From E 
as center, with radius E A 1 , describe the arc A 1 G indefinitely. 
Make the chord A 1 G equal to L K of the end elevation, Fig. 
405. Connect G E. Then A 1 E G will be a second section of the pattern. To this add E G II, equal to E A 1 F. 
Then FEHGA' will be the pattern sought. 

537. Patterns for a Soap-maker's Float. — Fig. 409 represents a soapmaker's float as commonly constructed 
in some places. The part A O B, or the bottom, is to be regarded as raised work, and shaped by means of the 
raising hammer without regard to any rules. The sides are to be considered as parts of two cones having ellip- 
tical bases, the short diameters of which are alike, but the long diameters of which vary. Tims in the plan, 
Fig. 410, L D 1 M represents the half of the base of an elliptical cone, the short diameter of which is equal to 




Fig. 407.— Diagram. 



Fig. 40S.— Pattern. 
Pattern of the Flaring End of an Oblong Pan. 
Second Case — When Top is Curved and Bottom 
is Straight. 



152 



Pattern Problems. 




Fig. 409.— Elevation. 
Soapmaker's Float. 



L M, and the half of the long diameter of which is equal to K 1 D 1 . By thus resolving the envelope of the 
vessel into sections of cones, the development of patterns becomes, comparatively speaking, a simple operation. 

First develop that part of the pattern which corresponds to A E F D of 
the section, Fig. 410. To do this proceed as follows : Drop the point A 
on to the center line of the plan, as shown by A 1 . As the curve D' L 
is the quarter of a perfect ellipse, it becomes necessary to locate the 
point P so that the curve A 1 P shall be a section of the same elliptical 
cone as D 1 L. This may be determined in this maimer : Connect D 1 L 
by a straight line, as shown. Then from A 1 draw a line parallel to D : L, 
cutting the short diameter, which point of intersection will be the point 
P required. "With this point determined, draw the curve A 1 P, being a 
portion of the regular ellipse, 
by any convenient method. 
Produce the line F E of the elevation in the direction of K indefi- 
nitely. In like manner produce D A of the elevation until it 
reaches F E produced in the point K. Then D K F may be re- 
garded as the section of a half cone, of which that part of the ves- 
sel indicated by A E F D is a portion, and K F the perpendicular 
bight. Next, divide both halves on the plan L D 1 M into the .same 
number of equal parts, as shown by the small figures 1, 2, 3, etc., 
running both ways from D'. Construct the diagram shown in Fig. 
412 by drawing the line D K? of indefinite length, and the line E 1 K 
at right angles to it, making K 1 K in length equal to F K, Fig. 410. 
Establish the point K? by making the distance K 1 E 2 equal to E F 
of Fig. 410. Draw E? A parallel to K 1 D. From each of the 
points 2, 3, 4, etc. of the plan, draw lines to the center K 1 , and set off 
distances equal to these lines upon the line K 1 D of Fig. 412, meas- 
uring from K 1 toward D. From each of the points thus obtained 
draw lines to the point K. cutting A E". With the foot of the com- 
passes in the point E, and the other brought successively against the 



points 1, 2, 3, etc., in the line D E 1 



H 




and the line A K ! 
producing 



describe arcs, 

them indeli- 

Take in the di- 

space equal to 




Fig. 410. — Plan and Inverted Section. 
Soapmaker's Float. 



the divisions 1, 2, 3, 4, etc., of Fig. 410, 



commencing at the point a in the first arc (Fig. 412), step to 



nitely. 
viders 
and, 

the second arc, and thence to the third arc, and thus continue 
stepping from one arc to another until the entire stretchout of the 
half plan has been laid off, as shown in Fig. 412. The same opera- 
tion is to be repeated upon the arcs drawn from the points in the 
line A K\ It may be shortened, however, by drawing radial lines 
to the point E from the several points determined in the first set of 
arcs. Then- a line ti^aced through the several points of intersection 
thus obtained, as shown by i c aud a d, will be the boundary lines 
of the pattern. The pattern for the other end of the article is to 
be, in the main, developed in the same manner as we have described. 
There is, however, a slightly different case arising, to which we shall 
give attention, without repeating that portion of the description 
which would coincide with what has been stated. The points P and 
B 1 being established, connect them by a true ellipse, half of the long 
diameter of which is E 1 B 1 , and half of the short diameter of which 
is K 1 P. It now becomes necessary to obtain a curve between the 
points L and C, which shall be a section of- the same elliptical cone as B 1 P. To do this proceed as follows: 
Connect the points P and B' by means of a straight line. From the point C draw a line parallel to P B 1 , and 



L b 'g 

Fig. 411.— Pattern for the Larger Half. 
Soapmaker's Float. 



Pattern Problems. 




D123J5G7 "«' 

Fig. 412.— Pattern for the Smaller Half. 
Soupmaker's Float. 



produce it until it cuts the line L G', which is a straight line drawn at right angles to L M. Then G 1 becomes 
a point in the lower base of the cone corresponding to the point P in the upper base. Draw the line G 1 P and 
continue it until it intersects the long diameter in H 1 . Drop the point G 1 vertical from the plan on to the base 
line D C of the elevation, as indicated by the point G. Draw a line through 
the points G and E, which produce indefinitely in the direction of H. In 
like manner produce the side C E of the cone until it intersects G E pro- 
duced in the point II. Then it will be found that the point H of the eleva- 
tion and the point IP of the plan coincide, as indicated by the line H IP. 
The operation of developing the pattern from this stage forward is the same 
as in the previous case, save only in the matter of the triangular piece indi- 
cated by G E F of the elevation. After completing the other portions of 
the pattern, this triangular piece is added as follows : The distance II 1 L in 
Fig. 411 is to be set off on the line H 1 C in the same manner as the other 
points — i. e., IF L of Fig. 411 is equal to IF L of Fig. 410. Then L is to 
be treated in the same manner as the other point, an arc being struck from 
it, as indicated in the engraving, by which to determine the corresponding 
point L = in the outline of the pattern. If G~ becomes equal to L G 1 of the 
plan, Fig. 410. From V, Fig. 411, draw a line to E. Then E L 2 G 2 will 
be the pattern of the triangular piece indicated in Fig. 410 by E F G. It 
is to be added upon the opposite end of the pattern in like manner, as 
indicated by E 1 G 1 L 1 . 

538. A Square Return Miter, or a Miter at Right Angles, as in a Cornice at the Comer of a Building. 
— In Fig. 413, let ABDC represent a cornice at the corner of the building for which a miter at right angles 

is desired. As has been elsewhere explained, the process of cutting a 
miter, when applied to a right angle, admits of certain abbreviations not 
employed in the use of other angles. The demonstration here introduced 
is calculated to show the method of obtaining the pattern for a square 
miter with the least possible labor. Divide the profile A B into any con 
venient number of parts, as shown by the small figures. At right angles 
to the lines of the molding, and in convenient proximity to it, lay off the 
stretchout E F, through the points in which, parallel to the lines of the 
cornice, draw measuring lines in the usual manner, producing them far 
enough to intercept lines dropped vertically from points in A B. Place 
the T-square at right angles to the cornice, or, what is the same, parallel 
to the stretchout line, and, bringing it successively against points in the 
profile A B, cut measuring lines of corresponding numbers. Then a line 
traced through these points, as shown by G II, will be the pattern sought. 
539. A Return Miter at Other than a Right Angle, as in a Cornice 
at the Corner of a Building. — In Fig. 414. let A B C D be a section of 
the cornice, of which a pattern is to be cut forming a miter in the angle, 
shown in plan by G H K. As remarked in the previous demonstra- 
tion, the cutting of a miter at other than a right angle demands 
certain work not necessary in the case of a right angle ; therefore 
the demonstration which follows is applicable in all cases save that 
of a right angle. Construct a plan of the required miter, as indicated 
by E F L K II G, and draw the miter line F II. By inspection it will 
be evident that F II is in reality the only line in the plan constructed 
which is used, therefore the work may be abbreviated to the extent of 
laying off simply the line F H, its inclination being determined by any means most convenient. The full 
plan, however is here introduced in order to show the requirements of that line. It must be drawn as though 
the complete plan were represented. Divide the profile A B in the usual manner into any convenient number 
of parts, and from the points thus obtained drop lines vertically on to the miter line in the plan F II, as shown. 
At right angles to one arm of the cornice, as shown in plan — in this case at right angles to E F — lay off a 




Fig. 4T3. — A Square Return Miter, or a 
Miter at Right Angles, as in a Cornice 
at the Corner of a Building. 



154 



Pattern Problems. 




stretchout of the profile, as shown by N M, through the points in which draw the usual measuring lines, as indi- 
cated. Place the T-square parallel to this line, or, what is the same, at right angles to E T, and, bringing it suc- 

, 3 A cessivelv against the points F II, cut measuring lines of corresponding 

numbers. Then a line traced through the points thus obtained, as shown 
by P, will be the pattern sought. It is evident that the stretchout M N 
could with equal propriety be laid off at right angles to F L, the general 
rule in miter cutting being that the stretchout must be laid off at right 
angles to the molding the pattern of which is being produced. By the 
operation shown above, MOPN represents a pattern of a portion of the 
molding shown in plan by E F H G. If for any reason the stretchout 

had been laid off at right angles to 
F L, the pattern produced would have 
represented a portion of the molding 
shown in plan by F L K IT. But since 
these two pieces are alike, all necessary 
results are accomplished in performing 
the operation once, and therefore it is 
performed at such a place as is most 
convenient, which, of course, is where 
the T-square can be used from adjacent 
sides of the board. 

539. A Butt Miter against a Plain 
Surface shown in Elevation. — Tet 
C D in Fig. 415 be the profile of a 
cornice, and A B the angle or inclina- 
tion of the surface in elevation against 
which the cornice miters. Tet AKLB 



414. — .4. Return Miter at Other than a Right Angle, as in a Cm-nice at the 
Corner of a Building. 



be the length of the cornice for which 



the pattern is desired. Space the profile in the usual manner, and from 
the points draw lines cutting the miter line A B. At right angles to 
the cornice lay off, on any convenient line, as E F, a stretchout of the 
profile C D, through the points in which draw the usual measuring 
lines, all as indicated by the small figures. Placing the T-square at 
right angles to the lines of the cornice, or, what is the same, parallel 
to the stretchout line, bring it successively against the points in the 
miter line A B and cut corresponding measuring lines, as indicated by 
the dotted lines. A line traced through these points, as indicated by 
IT G, will be the pattern required. 

540. A Butt Miter against a Regular Curved Surface. — In Fig. 
416, let A B be the profile of any cornice, a butt miter in which is to 
be cut to fit it against a surface, the profile of which is a regular curve, 
as shown by C D. Space the profile in the usual manner, and through 
the points draw lines cutting C D. At right angles to the line of cor- 
nice lay off the stretchout T M, as shown, through the points in which 
draw measuring lines in the usual manner. Place the T-square parallel 
to the stretchout line, or, what is the same, at right angles to the lines 
of the cornice, and, bringing it against the several points in C D, cut 
the corresponding measuring lines, as shown. In the event of a wide 
space, as shown by a 1 V in the elevation, two methods are at the choice 
of the pattern cutter. One is to divide this space in the profile in the 
usual manner, as though it was a molding from which to obtain a number of points approximating to the curve. 
The other method is as given in the engraving. Transfer to the pattern G 1 a point corresponding to G of 




Fig. 415. 



—A Butt Miter against a Plain Sur- 
face shown in Elevation. 



Pattern Problems. 



155 




M H' 

Fig. 416. — A Butt Miter against a Regular 
Carved Surface. 



the elevation, the center by which the curve C D was struck, as indicated by the line and arrow point. Then 
from G 1 as center, with the same radius as used to strike the curve in the elevation, strike the arc a b, extending 
from the measuring line 11 to measuring line 12. A line traced through the several points of intersection 
together with the arc struck from the center G 1 , as above explained, all 
as shown by E F, will be the shape of the required pattern. 

511. A Putt Miter against a Plain Surface shown in Plan. — 
Let C D in Fig. 417 be the profile of the cornice which is required to 
miter against a vertical surface standing at any angle with the lines of 
the cornice, the angle being shown in plan by A B. Draw the profile 
C E>, corresponding to the lines of the cornice, all as indicated. Space 
in the usual manner, and through the points draw lines cutting the 
miter line A B. At any convenient point at right angles to the lines 
of the cornice, lay off the stretchout E F of the profile C D, through 
the points in which draw measuring lines in the usual manner. Plac- 
ing the T-scpiare at right angles to the cornice, or, what is the same, 
parallel to the stretchout line E F, bring it successively against the 
points in A B and cut the corresponding measuring lines. A line 
traced through the points of intersection thus obtained, shown by K G, 
will be the pattern required. 

542. A Butt Miter of a Molding Inclined in Elevation against 
a Plain Surface Oblique in Plan. — Let A B in Fig. 418 be the pro- 
file of a given cornice, and let E D C F represent the rake or incline of 
the" cornice as seen in elevation. Let G II represent the angle of the 
intersecting surface in plan. The first step in developing the pattern 
will be to obtain miter lines in the elevation, as shown by E F. For this purpose draw the profile A B in con- 
nection with the raking cornice, which space in the usual manner, as indicated by the small figures. Draw a 
duplicate of this profile, as shown by A 1 B', placing it in a horizontal position, with points corresponding to 
those shown in the raking cornice. Space the profile A 1 B 1 into the same number of parts as A B, and through 

the points thus obtained carry 
_!4_,i3 lines parallel to the lines of the 

cornice, as seen in plan, cutting 
the miter line G IT, as shown. In 
like manner draw lines through 
the points in A B, carrying them 
parallel to the lines of the raking 
cornice in the direction of E F 
indefinitely, as shown. Place the 
T-square at right angles to the 
lines of the cornice, as shown 
in plan, and, bringing it against 
the points of intersection in 
the line G IT, carry lines ver- 
tically, cutting corresponding 
lines in the inclined cornice 
drawn from the profile A B. 
Through the points of intersec- 
tion thus obtained trace a line, 
as shown from E to F. Then 
this profile E F will be the miter 



h-H 

M il 



1 D 




9 10 



! ! B 



s 



-H-f 



tz: 



Fig. 417. — A Butt Miter against a Plain Surface shown in Plan. 



line in elevation, formed by a cornice of the profile A B meeting a surface in the angle shown by G II in the 
plan. At right angles to the raking cornice lay off a stretchout upon any line, as K L, and through the points 
draw the usual measuring lines, all as shown. Place the T-square at right angles to the lines of the raking cor- 
nice, and, bringing it against the several points in the profile E F, cut corresponding measuring lines drawn 



156 



Pattern Problems. 



from the stretchout K L. A line traced through these points of intersection, as shown from M to N", will be 

the pattern required. 

543. A Butt Miter against an Irregular or Molded Surface. — Let B A in Fig. 419 be the profile of a cor- 
nice, against which a molding of the pro- 
file, shown by G H, is to miter, the latter 
meeting it at an angle, as indicated by C D. 
Draw the profile B A ; also construct an ele- 
vation of the cornice meeting it, as shown 
by C D F E, in line with which draw the 
profile G II. Divide G H in the usual man- 
ner into any number of convenient parts, 
and through the points draw lines par- 
allel to the lines of the inclined molding, 
cutting the profile B A, all as indicated by 
the dotted lines. At right angles to the 
lines of the inclined molding lay off a 
stretchout, M N, in the usual manner, 
through the points in which draw measur- 
ing lines. Place the T-square at right angles 
to the lines of the inclined molding, or, what 
is the same, parallel to the stretchout line, 
and, bringing it against the points of inter- 
section formed by the lines drawn from the 
profile G II across the profile B A, cut the 
corresponding measuring lines. In the event 
of any angles or points occurring in the pro- 
file B A which are not met by lines drawn 
from the points in G H, additional lines 
from these points must be drawn, cutting 
the profile G H, in order to establish corre- 

N 




Fig. 418. — A Butt Miter of a Molding Inclined in Elevation against a 
Plain Surface Oblique in v lan. 

sponding points in the stretchout. Thus the points 3 and 13 in the 
profile G H are inserted after spacing the profile, as above described, 
because the points with which they correspond in the profile B E 
are angles which must be clearly indicated in the pattern to be cut. 
Having thus cut the measuring lines corresponding to the points in 
the profile B A, draw a line through the points of intersection, as 
shown by P. Then P will be the shape of the pattern of the 
incline cornice to miter against the profile A B. 

544. Miter between Two Moldings of Different Profiles. — To construct a square miter between moldings 
of dissimilar profiles requires two distinct operations. The miter upon each piece is to be cut as it would 
appear when intersected by the other molding. Let the profiles A B and A 1 B 1 in Figs. 420 and 421 be of the 




Fig. 419. — A Butt Miter against an Irregular or 
Molded Surface. 



Pattern Problems. 



157 





F 


C .A 


1 


. £. 


hv~ 


\^ 


vA ■ 


-^L. 








V" 




Jj 


1 || I c 


t 


) B 

c' 




I 






1 








4 

5 

7 
it 




L::r\ 












! i|i! 












! \0 


9 




/ 


10 
12 




/ 




/ 




V 


IJ 




1 


'4 


i 


E 1 


1 


1 


D 




Fig. 420.— First Operation. Fig. 421.— Second Operation. 

Miter between T-wo Moldings of Different Profiles. 



same liiglit, but differing in members, between which a square miter is to be formed. Proceed as follows : 
Draw E F, a duplicate of A 1 B 1 , in line with A B. Divide A B into any convenient mvrnber of parts 
in the usual manner, from which carry lines horizontally against E F, and thereby construct an elevation 
of the molding as it would ap- 
pear if intersected by F E, all 
as shown by F C D E. For 
the pattern of this piece, at 
right angles to its lines lay off 
a stretchout, G II, of the pro- 
file A B, through the points 
in which draw the usual meas- 
uring lines. Bring the T-square 
against the points of intersec- 
tion in the line E F, and cut 
the corresponding measuring 
lines. Then a line traced 
through these points, as shown 
by E 1 F', will give the shape 
of the cut to fit the molding 
against the profile E F. For the other piece proceed in the same manner, reversing the order of the profiles. 
Draw M N", a duplicate of A B, in line with A 1 B 1 . Divide A 1 B 1 in the usual manner. Through the 
points draw lines cutting M 1ST, thereby constructing an elevation, K M 1ST L, of the piece the pattern of which is 
sought. At right angles to this piece lay off the stretchout P of the profile A 1 B 1 , through the points in 

which draw measuring 
lines, as shown. "With 
the T-square at right an- 
gles to the lines K M 1ST L, 
and brought against the 
points in M 1ST, cut corre- 
sponding measuring lines 
drawn through O P. 
A line traced through 
these points, as shown 
by WW, will be the 
shape of the piece re- 
quired to fit against the 
profile M K In the 
event of the points ob- 
tained by spacing the 
profiles A B and A 1 B' 
not meeting all the points 
in the profiles F E and 
M N" necessary to be 
marked in the pattern, 
then lines must be drawn 
backward from such 
points in profiles M 1ST 
and E F, cutting the pro- 
file A' B' or A B, as 
the case may be. Cor- 




Fig. 422 



-The Patterns of the Moldings bounding a Panel, the Shape of which is a Scalene 

Triangle. 

responding points are then to be inserted in the stretchouts, through which measuring lines are to be drawn, 
which in turn are to be intersected by lines dropped from the points. An illustration of this occurs in 
point No. 6£ in Fig. 421. It will be seen that this point is absolutely essential to the shape of the pattern. 



158 



Pattern Problems. 



Therefore, after spacing the profile a line is drawn from X back to A 1 B', forming the point jSTo. 6£. In turn 
this point is transferred to the stretchout O P, also marked 6J, from which a measuring line is drawn in the 
same manner as through the other points in the stretchout, upon which a point from X is dropped, as shown by 
X 1 . In actual practice such expedients as this must be resorted to in almost every case, because usually there is 
less correspondence between the members of dissimilar profiles, between which a miter is required, than in the 
illustration here given. By this means profiles, however unlike, can be joined. 

545. The Patterns of the Moldings bounding a Panel, the Shape of which is a Scalene Triangle. — In Fig. 
422, let D E F be the elevation of a triangular panel or other article, surrounding which is a molding of a pro- 
file, shown at G and G 1 . Construct an ele- 
vation of the panel, as shown by ABC, 
and draw the miter lines A D, B E, C F. 
For the patterns of the several sides pro- 
ceed as follows : Draw a profile, G, placing 
it, relative to the side D F, in the position 
corresponding to the molding to be con- 
structed. Divide it into any convenient 
number of parts in the usual manner, and 
through these points draw lines, as shown, 
cutting the miter lines F C and A D. In 
like manner place the profile G 1 in a corre- 
sponding position. Divide it into the same 
number of parts, and draw lines intersect- 
ing those drawn from the first profile in the 
line F to C, also cutting the line E B. By 
this operation we have points in the three 
miter lines A D, E B, F C, from which to 
lay off the pattern in the usual manner. At 
right angles to each of the three sides, at 
convenient points, draw stretchout lines, as 
shown by IT I, H 1 1" and H 2 F, through the 
points in which draw the usual measuring 
lines. "With the T-square parallel to each 
of the several stretchout lines, or, what is 
the same, at right angles to the respective 
sides, bringing the blade successively against 
the points in the several miter lines, cut 
the corresponding measuring lines, all as 
indicated by the dotted lines. Then lines 
traced through the points of intersection 
thus obtained will describe the patterns 
required. A 1 C 1 F 1 D 1 will be the pattern 
for the side, A D F C of the elevation, 
and likewise C 2 B 2 E 2 F 2 is the pattern for 
the side, described by similar letters. 

546. A Face Miter, or Miter at Right Angles, as in the Molding Around a Panel. — In Fig. 423, let 
A B C D represent any panel, around which a molding is to be carried of the profile E and E'. The miters 
required in this case are of the nature commonly known as "face" miters, which in the process of pat- 
tern cutting require substantially the same steps as indicated in the preceding problem for a miter at any angle 
other than a right angle around a panel. That is to say, by reason of the position in which the profile is 
shown, it is necessary to drop points against a miter fine, and thence carry them to the measuring lines, in 
order to develop the pattern. For the patterns, therefore, we proceed as follows : Draw profiles in opposite 
sides of the panel, as shown by E and E 1 , or, what is the same, draw a section of the panel as is shown by the 
lines across its width. Divide the two profiles in the usual manner into the same number of parts. Through 




Fig. 423. — A Face Miter, or Miter at Right Angles, as in the Molding 
Around a Panel. 



Pattern, Problems. 



159 



the angles of the panel draw miter lines, as shown by A F and C G. From the points in the profile already 
determined, draw lines parallel to the lines of the molding, cutting these miter lines, as shown. For the pattern 
of the side corresponding to A B, lay off a stretchout at right angles to it, as shown by H K, through which 
draw measuring lines in the usual manner. Place the T-square at right angles to A B, or, what is the same, 
jDarallel to the stretchout line II K, and, bringing it successively against the several points in the miter line A F, 
cut measuring lines of corresponding number. Then a line traced through these points, as shown by L M, will 
be the pattern sought. In like manner, by dropping points from the profile E 2 on to the miter line C G, the 
pattern for the opposite side may be obtained, all as shown by L 1 IF. So far as the molding bounding the panel 
is concerned, these two patterns correspond in all particulars, the only difference being that an allowance is 
made for a seam in connection with the lower piece, whereas a fiat surface to form the panel itself is shown 
attached to the upper piece. The two patterns are presented, in order to show the convenience of working 
from both sides in pro- 
ducing the two pieces, 
instead of copying one 
from the other. The 
pattern of the end piece 
is derived from the two 
miter lines A F and C G, 
from the points already 
established in them. It 
is quite as easy to describe 
the pattern by this means 
as to copy it from the 
pattern first obtained. 
For the pattern of the 
end piece, at right angles 
to the end of the j^auel 
A C, lay off a stretchout 
of the profile, as shown 
by N O, through the 
points in which draw 
measuring lines in the 
usual manner, producing 
them sufficiently far in 
each direction to inter- 
cept lines dropped from 
the points in the two 
miter lines. Place the 
T-square at right angles 




Fig. 424. — The Patterns of a Molding Mitering Around an Irregular Four-sided Figure. 



to A C, and, bringing it successively against points in A F and C G, cut measuring lines of corresponding 
numbers. Then lines traced through the intersections thus formed, as shown by P B and S T, will be the 
shape of the pattern of the end piece. 

547. The Patterns of a Holding Mitering Around an Irregular Four-sided Figure. — In Fig. 424, let 
A B C D be the elevation of an irregular four-sided figure, to which a molding is to be fitted of the profile 
shown by K and K 1 . Place the profile in two of the sides, as shown, and construct an elevation of the molding 
as it would appear when finished, as shown by E F G II. Draw the several miter lines B F, C G, D H and A E. 
Divide the two profiles into the same number of j>arts in the usual manner, through the points in which draw 
lines parallel to the lines of the molding in which they occur, cutting the miter lines, as shown. At right 
angles to each of the several sides lay off a stretchout from the profile, as shown by L M, L 1 M', L 2 M, L 3 M\ 
Through the several points in these several stretchouts draw measuring lines in the usual manner, producing 
them until they are equal in length to the respective sides, the pattern of which is to be cut. Placing the 
T-square at right angles to the lines of the several sides, or, what is the same, parallel to the stretchout lines, 
bring it against the points in the miter lines, cutting the corresponding measuring lines, all as indicated by the 



160 



Pattern Problems. 



dotted lines. Then the lines traced through these points of intersection will give the several patterns required. 
Thus E 1 II' D 1 A 1 will be the pattern of the side E II D A of the elevation, and IP D 5 C 1 G 3 will be the pat- 
tern of the side II DC6, and so on for the others. 

548. The Patterns of Simple Gable Miters— In Fig. 425, let A B K It be the elevation of the miters of 
a cornice at the foot and peak of a gable. The conditions of the elevation are established by the requirements 
of the work. Let II be the profile of the molding, as shown at the extremity of the horizontal part. Draw 
the miter line B C, separating the horizontal part from the raking part and the miter line K L at the top. 
Divide the profile H in the usual manner into any convenient number of equal parts. Place the T-square par- 
allel to the lines in the horizontal molding, and, bringing it successively against the points in the profile, cut 

the miter line B C, as shown. At 
right angles to the lines of the hor- 
izontal cornice draw the stretchout 
E F, through the points in which 
draw the usual measuring lines, as 
shown. Reverse the T-square, let- 
ting the blade lie parallel to the 
stretchout Hne E F, and, bringing 
it against the several points of the 
profile Pi, cut the corresponding 
measuring lines. Then a line traced 
through these points of intersection, 
as shown from G to ~Y, will be the 
pattern of the end of the horizontal 
cornice mitering with the return. 
In like manner, with the T _s q uare 
in the same position, bring it against 
the points in the miter line B C, and 
cut the corresponding measuring 
lines drawn through the stretchout 
E F. Then a line traced through 
the points of intersection thus ob- 
tained, as shown by T U, will be 
the pattern of the end of the hori- 
zontal cornice mitering against the 




raking cornice. 



At right angles to 



The Patterns of Simple Gable Miters. 



the lines of the raking cornice draw 
a duplicate profile, as shown by IP 1 , 
which divide into any convenient 
number of equal parts, all as indi- 
cated by the small figures. Through 
these points draw lines cutting the 
miter line B C, and also the miter 
line K L at the top. At right angles to the lines of the raking cornice draw the stretchout line E 1 F 1 equal to 
the profile IF, through the points in which draw the usual measuring lines, as shown. Place the T-square par- 
allel to this stretchout line, and, bringing it successively against the points in B C and K L, cut the correspond- 
ing measuring lines, all as indicated by the dotted lines. Through the points thus obtained trace lines, as indi- 
cated by IT X and O P. Then M 1ST will be the pattern for the bottom of the raking cornice mitering against 
the horizontal, and O P will be the pattern for the top of the raking cornice. The pattern shown at G Y will 
also be the pattern for the return mitering against A D of the elevation, it being necessary only to establish its 
length, which may be done from a plan drawn in connection with the elevation or from actual measurements of 
the work. 

549. To Ascertain the Profile of a Horizontal Molding Adapted to Miter with a Given Inclined Molding 
at Eight Angles in Plan, and the Several Miter Patterns Involved.— In the elevation B C E D, and plan 



Pattern Problems. 



161 



G H K, of Fig. 426, is presented one of the sets of conditions which necessitate a change of profile, in either 
the horizontal or raking molding, in order to accomplish a miter joint at the point indicated by I II in the plan. 
In other words, the conditions are such that with a given profile, as shown by A' in the raking molding, the 
horizontal molding forming the return will require to be modified, as shown by the profile A 3 , in order to form 
a miter upon the line I H in the plan ; or, if A 3 is established, A' will have to be constructed to correspond with 
A 2 . The reason for this is quite obvious. The distance across the raking molding at right angles to its lines is 
greater than the corresponding distance across the return molding at right angles to its lines ; therefore the 
projection in the cornice, as shown by the profile A\ must be distributed through a smaller space than is shown 
in the profile A 1 . In this problem we assume that the pitch of the raking cornice B C is established and that 
the profile A is given, and from 
these parts it is required to de- J 
velop the modified profile. We 
have the choice of placing the 
normal profile in the horizontal 
return and making the raking 
profile correspond with it, or of 
placing the normal profile in the 
raking molding and making the 
profile of the horizontal mold- 
ing agree with it. Although 
the principle upon which these 
ojierations is performed is iden- 
tical in both, the demonstration 
will be made clearer if each is 
fully illustrated independent of 
the other. In this problem and 
the following one, therefore, we 
show the several steps necessary 
to take in modifying the profile, 
and in cutting the several pat- 
terns required to form the struc- 
ture indicated by the elevation 
and plan. First we will assume 
that the normal profile occurs in 
the raking cornice, and that the 
horizontal profile is to be modi- 
fied to suit it. We then proceed as 
follows : Draw a representation 
of the normal profile in the rak- 
ing cornice, as shown by A 1 , plac- 
ing it to correspond to the lines 
of the cornice, as shown. Draw 
another profile corresponding to 
it in all parts, directly above or 
below the foot of the raking cornice, in line with the face of the new profile to be constructed, placing this pro- 
file A so that it shall correspond with the lines of the horizontal cornice. Divide the profiles A and A' into the 
same number of parts, and through the points thus obtained draw lines, those from A 1 being parallel to the lines 
of the raking cornice, and those from A intersecting them vertically. Through these points of intersection 
trace a fine, which gives the modified profile, as shown by A 5 . Then A 3 is the profile of the horizontal return, 
indicated by G II I F in the plan. It is also the elevation of the miter line I II of the plan for the several 
patterns involved. We therefore proceed as follows : At any convenient point at right angles to the lines 
of the raking cornice lay off the stretchout M 1ST of the profile A 1 , through the points in which draw 
measuring lines in the usual manner. Place the T-square at right angles to the lines of the raking cornice, and, 




Fig. 426. — To Ascertain the Profile of a Horizontal Molding Adapted to Miter ivith a Given 
Inclined Molding at Right Angles in Plan, and the Several Miter Patterns Involved. 



162 



Pattern Problems. 



bringing it successively against the points in the profile A", cut the corresponding measuring lines just described. 
Through the points of intersection trace a line, as shown by O P K. Then P R will be the shape of the lower 
end of the raking cornice inhering against the return. For the pattern of the return proceed as follows. Con- 
struct a side elevation of the return, as shown by S V U T, making the profile V TJ to correspond to the pro- 
file A" of the elevation, all as shown by B D. Let the length of the return correspond to the return as shown 
in the plan F G I. In the profile V U set off points corresponding to the points in the profile A 2 , as shown 
from B to T>. At right angles to the elevation of the return lay off a stretchout of Y TJ, or, what is the same, 

of the profile A 2 , as shown by W X, 
through the points in which draw 
measuring lines in the usual man- 
ner. Placing the T-square parallel 
to this stretchout line, and bringing 
it successively against the points in 
V U, cut the corresponding meas- 
uring lines. Then a line traced 
through these points of intersec- 
tion, as usual, from T to Z, will be 
the pattern of the horizontal return. 
550. Prom a Given Horizontal 
Molding, to Establish the Profile 
of a Corresponding Inclined 
Holding to Miter with it at Bight 
Angles in Plan, and the Several 
Miter Patterns hivolved. — The 
conditions shown in this problem 
are similar to those in the one just 
demonstrated. In this, however, 
the normal profile is given to the 
horizontal return, and the profile or 
the raking cornice is modified to 
correspond with it. To obtain the 
new profile we proceed as follows : 
Divide the normal profile A 1 , Fig. 
426, into any convenient number of 
parts in the usual manner, and from 
these points carry lines parallel to 
the lines of the raking cornice in- 
definitely. At any convenient point 
outside of the raking cornice, and 
at right angles to its lines, construct 
a duplicate of the normal profile, 
as shown by A", which divide into 
like number of spaces. With the 
"[■-square at right angles to the lines 
of the raking cornice, and brought 
successively against the several 
points in this profile, cut corresponding lines drawn through the cornice from the profile A 1 . Then a line traced 
through these points of intersection, as shown by A 3 , will be the profile of the raking cornice. For the pattern 
of the foot of the raking cornice mitering against the return, take the stretchout of the profile A 3 and lay it off 
on any line at right angles to the raking cornice, as shown by P O. Through the points in this stretchout line 
draw the usual measuring lines, as shown. With the J-square at right angles to the lines of the raking cornice, 
or, what is the same, parallel to the stretchout line, and, bringing it successively against the points in the profile 
A 1 , which is also an elevation of the miter, cut the measuring lines drawn through the stretchout P O. Then a 




Fig. 427. — From a Given Horizontal Molding, to Establish the Profile of a Correspond- 
ing Inclined Molding to Miter with it at Bight Angles in Plan, and the Several Miter 
Patterns Involved. 



Pattern Problems. 



163 



line traced through the points of intersection, as shown by B' E, 1 , will be the miter pattern of the foot of the 
raking cornice. For the pattern of the return proceed as follows : Construct an elevation of the return, as shown 
by F 1 G 1 K 1 II 1 , in dimensions making it correspond to F6K II of the plan. Space the profile A of the 
elevation of the return in the same manner as A'. At right angles to the lines in the return cornice draw any- 
straight line, as M N, on which lay off a stretchout of the profile A, through the points in which draw measuring 
lines in the usual manner. With the J-square at right angles to the lines of the return cornice, and bringing 
it successively against the points 
ofile A. cut the 



in tb 



pre 



sponding measuring lines, 
line 



Through 



corre- 
In like 
inner draw a line correspond- 
to F 1 IF of the side elevation, 
the points of intersec- 
tion obtained from the profile trace 
a line, as shown by G 2 E?. Then 
F 2 G 2 K 2 IF will be the pattern of 
the horizontal return to miter with 
the raking cornice, as described. 

551. In a Broken Pediment to 
Ascertain the Profile of the Hori- 
zontal Return at the Toji, Together 
with its Miters. — Still another 
change of profile in connection 
with gable and pediment cornices 
occurs in constructions commonly 
known as "broken pediments." 
Whether the normal profile be 
placed in the horizontal return at 
the foot of the gable or in the rak- 
ing cornice, a third profile is to be 
constructed by which to cut the 
patterns and establish the shape of 
the return occurring at the top. 
In Fig. 428, CBD represents a 
section of a broken pediment, of 
which the normal profile is A 1 . 
The profile for the return cornice 
at the foot of the gable, as shown 
by B C, is to be obtained by the 
rule just explained in Fig. 427. 
The profile for the return at the 
top of the raking cornice, as shown 
by A 2 , is to be obtained in the fol- 



lowing manner 



Divide the profile 
A 1 of the raking cornice into any 
convenient number of parts in the 
usual manner, and through these 




Fig. 428. — In a Broken Pediment to Ascertain the Profile of the Horizontal Return 
at the Top, Together with its Miters. 



points draw lines parallel to the lines of the cornice indefinitely. At any convenient point outside of the cor- 
nice, and in a vertical line with the point at which the new profile is to be constructed, draw a duplicate of the 
profile of the raking cornice, as shown by A, which space into the same number of parts as A.', already described. 
From the points in A draw lines vertically, intersecting lines drawn from A 1 . Then a line traced through these 
several points of intersection, as shown by A 2 , will constitute the profile of the horizontal return at the top and 
also the miter line as shown in elevation. As before remarked, it matters not whether the normal profile occurs 
in a horizontal cornice at the base or in a raking cornice, a change still remains to be made at the top. In which- 



164 



Pattern Problems. 



ever way the profile occurs the steps to be taken are the same as above described. If the normal profile were 
in the horizontal cornice at the foot of the gable and the modified profile in the position of A 1 , it would be 
immaterial whether the normal profile or a duplicate of the modified profile were in the place of A by which 

to obtain the intersecting lines. It 
is obvious that it can make no 
difference which is employed, 
for what we have to deal with is 
the projection only, which is the 
same in both cases. In this con- 
nection it may be remarked that 
the normal profile may be located 
in the horizontal return at the top 
of the other profiles, established by 
working from or reversing the sev- 
eral steps here described. "We are 
led to allude to the possible modifi- 
cations of the plan here suggested 
by reason of the demands made 
upon pattern cutters in cornice 
work, owing to the whimsicalities 
of modern designers. For the pat- 
terns of the several parts shown in 
the elevation proceed as follows: 
At right angles to the lines of the 
raking cornice lay off a stretchout 
of the profile of the raking cornice 
A, as shown by F G, through the 
points in which draw measuring 
lines in the usual manner. Place 
the T-square at right angles to the 
lines of the raking cornice, and, 
bringing the blade successively 
against the points in the profile A 2 , 
which is also the miter line in the 
elevation, cut the corresponding 
measuring lines, and through these 
points of intersection trace a line, 
as shown by G H. Then G H will 
be the pattern of the top of the 
raking cornice to miter against the 
horizontal return. For the hori- 
zontal return the usual method 
would be to construct an elevation 
of it in a manner similar to that 
described for the return at the foot 
of the gable in the preceding dem- 
onstrations; the equivalent of this, 
however, can be done in a way 
of the labor. Draw the line K M perpendicular to the lines of the 
be if shown in elevation. Upon K M lay off a stretchout of the profile 
figures, and through the points draw the usual measuring lines. "With the 




Fig. 42g. — From a Given. Horizontal Profile, to Establish the Profile for a Corresponding 
Inclined Molding to Miter with it at an Octagon Angle in Plan, and the Several 
Miter Patterns Involved. 



to save a considerable portion 
horizontal return, as it would 
A", all as shown by the small 
T-square parallel to the stretchout line K M, 



the blade successively against the points in the profile A s , 
which is also the miter line in elevation, cutting the corresponding measuring lines. Through these points of 



Pattern Problems. 



165 



a Corresponding Inclined Molding to 



intersection trace a line, as shown by 1ST L, which will he the pattern of the end of the horizontal return to miter 
against the gable cornice, as shown. 

552. From a Given Horizontal Profile, to Establish the Profile for 
Miter with it at an Octagon Angle in Plan, and the Several Miter Pat- 
terns Involved. — Another example wherein is required a change of pro- 
file in order to produce a miter between the parts is shown in Fig. 429. 
In this case the angle shown in plan between the abutting members is 
that of an octagon, as indicated by BCD. To produce the modified pro- 
file and to describe the patterns we proceed as follows : In the side B C 
draw the profile A, as indicated, and in the corresponding side, as shown in 
elevation by N O L K, draw a 

duplicate profile, as shown by A'. y >=i T 

Divide both of these profiles into 
the same number of parts, and 
from the points carry lines par- 
allel to the lines of molding in 
the respective views. Then pro- 
duce the hues drawn from A 
until they meet the miter line 
C X. From the points thus ob- 
tained in C X carry lines verti- 
cally until they meet those drawn 
through A 1 , intersecting in points 
as shown from L. Through 
these points of intersection draw 
the line O L, which will be 
the miter line in elevation cor- 
responding to the miter line 
C X of the plan. From the 
points in O L carry lines up the 
rating molding in the direction 
of P M indefinitely. At any 
convenient point outside of the 
raking cornice draw a duplicate 
of the normal profile, as shown 
by A 2 , placing its vertical line at 
right angles to the lines of the 
raking cornice. Divide the pro- 
file A 2 into the same number of 
spaces as employed in A and A 1 , 
and from these points carry lines 
at right angles to the lines of the 
raking cornice, intersecting those 
drawn from the points in L. 
Trace a line through these inter- 
sections, as shown from II to S. 
Then B S will be the required profile of a raking cornice to miter against a level cornice of the profile 
A at an angle indicated by C X in the plan, or an octagon angle. For the pattern of the level cornice, 
at right angles to the arm B C in the plan lay off a stretchout of the profile A, as shown by E F, 
through the points in which draw the usual measuring line. "With the T-square at right angles to B C, 
bringing the blade successively against the several points in X C, cut corresponding measuring lines drawn 
through E F. Then a line traced through these points, as shown from H to G, will be the required 
pattern of the horizontal cornice. In like manner, for the pattern of the raking cornice, at 




Fig. 430. — From a Given Profile in an Inclined Molding, to Establish the Profile of a Cor- 
responding Horizontal Molding to Miter with it at an Octagon Angle in Plan, and the 
Miter Patterns Involved. 



right 



angles 



166 



Pattern Problems. 



to its lines lay off a stretchout of the profile B S, as shown by TJ T, through the points in which draw 
measuring lines in the usual manner. With the T-sq uare a f right angles to the lines of the raking cornice, 
and brought successively against the points in the miter line O L, as shown in elevation, cut the corre- 
sponding measuring lines. Then a line traced through the points thus obtained, as shown by ~W Y, will 
be the required pattern for the raking cornice. 

553. From a Given Profile in an Inclined Holding, to Establish the Profile of a Corresponding Horizon- 
tal Molding to Miter with it at an Octagon Angle in Plan, and tlie Miter Patterns Involved. — In Fig. 430, 
let B C D be the angle in plan at which the two sections are to join, and U O V the angle in elevation. To 

• form a miter between moldings meeting under these conditions a change of profile is required. To obtain the ■ 
modified profile and the miter line in elevation proceed as follows : Draw the normal stay A with its vertical 
side parallel to the lines in the plan of the arm corresponding to the front of the elevation, all as shown by 
E X D C. Draw a duplicate of the normal profile in correct position in the elevation, as shown by A 1 . Divide 
both of these stays into the same number of parts, and through the points draw lines parallel, in the one case 

with the lines in the plan and in 
the other with the lines of the rak- 
ing cornice, all as indicated by the 
dotted lines. From the points in 
the miter line of the plan C E, ob- 
tained by the lines drawn through 
the stay A, carry lines vertically 
intersecting the lines drawn from 
A 1 . Then a line traced through 
the intersections thus obtained, as 
shown from N to 0, will be the 
miter line in elevation. From the 
points in 1ST carry lines horizon- 
tally along the arm of the horizon- 
tal molding 1ST O U T, as shown. 
At any convenient point outside of 
this arm, either above or below it, 
draw a duplicate of the normal stay, 
as shown by A 2 , which divide into 
the same number of parts as before, 
and from the points carry lines ver- 
tically intersecting the lines drawn 
■from X O, just described. Then a 
line traced through these points of 
intersection, as shown by T S, will 
give the modified profile. For the 
patterns of the parts proceed as 
follows : For the pattern of the arm Y N O IT, at right angles to the same as shown in plan by "W E C B, lay 
off on any straight line, as G F, a stretchout of the profile T S, all as shown by the small figures V, 2 2 , 3 2 , etc. 
Through these points draw measuring lines in the usual manner. With the T-square parallel to the stretchout 
line, and brought against the points of the miter line E C in plan, cut corresponding measuring lines, as indi- 
cated by the dotted lines, and through these points of intersection trace a line, as shown by K H. Then K II 
will be the shape of the end of Y N" O II to miter against the raking molding. For the pattern of the raking 
molding, at right angles to the arm NZVO in the elevation lay out a stretchout, L M, from the profile A 1 . 
Through the points in this stretchout draw measuring lines in the usual manner. Place the J-square parallel to 
the stretchout line, or, what is the same, at right angles to the arm KZVO, and, bringing it against the several 
points in the miter line in elevation N" O, cut corresponding measuring lines, as indicated by the dotted lines. 
Then a line traced through these points of intersection, as shown by P P, will be the shape of the cut on the 
arm N Z Y O to miter against the horizontal molding. 

554. Patterns for the Moldings and Roof Pieces in the Gables of a Square Pinnacle. — Fig. 431 shows 




Fig. 431. — Patterns for the Moldings and Roof Pieces in the Gable of a Square 

Pinnacle. 



Pattern Problems. 



167 



the elevation of one of four similar gables occurring in a square pinnacle. The profile of the molding is shown 
by P. The first step is to obtain the miter line shown at K, from which to measure for the pattern. Draw the 
profile P in the molding, as shown, placing it so that its members will correspond with the lines of the mold 
ing. Draw a second profile, P l , in the side view of the gable, placing it, as shown in the engraving, so that 
its members will coincide with the line of the side view, and also with the first profile already drawn. Space 
both of these profiles into the same number of parts in the usual manner, and through the points thus obtained 
draw lines parallel to the lines of the elevation, as shown. Trace a line through these intersections. Then K 
is the line in elevation upon which the moldings will miter. Draw the miter line M for the top of the 
gable, as shown. Upon any line, as G H, drawn at right angles to the line of the gable in elevation, lay off a 
stretchout of the profile, as shown by the small figures. Through these points draw measuring lines, as shown. 
Place the T-square parallel to the stretchout line, or, what is the same, at right angles to the line of the gable, 
and, bringing it successively against the several points in O M and the miter line K, cut the corresponding 
measuring lines, as shown. Make E 1 D' 
equal to E D of the side view of the 
gable, and set it off at right angles 
to E 1 B 1 . In like manner, at right 
angles to the same line, set off A 1 B' 
equal to A B of the side view. Draw 
the line indicated by A 1 D 1 , as shown, 
and trace lines through the intersection 
of points dropped from the elevation on 
to the measuring lines, thus completing 
the patterns. 

555. The Pattern for the Miter 
Between the Holdings of Adjacent Ga- 
bles Upon a Square Shaft, Formed by 
Means of a Ball.— In Fig. 432, let A C 
be one of the gables in profile and B D 
the other in elevation, the moldings form- 
ing a joint against a ball, the center of 
which is shown at E. For the patterns 
we proceed as follows : Place the profile 
in each gable as shown by F and F 1 , 
locating them in such a manner with 
regard to their respective positions that 
corresponding points in each shall fall 
upon the same lines. Divide each of 
these profiles into the same number of 
equal parts, as indicated by the small 
figures. From the points thus obtained 




Fig. 432.— The Pattern for the Miter Between the Moldings of Adjacent Gables 
Upon a Square Shaft, Formed by Means of a Ball. 



in F drop lines vertically, meeting the profile of the ball, as shown from C to F. From the center E of the 
. ball erect a vertical line, as shown by E F. From the points in C F already obtained carry lines horizontally, 
cutting E F, as shown, and. thence continue them, by arcs struck from E as center, until they meet cor- 
responding points dropped from the profile F 1 by lines parallel to the gable in elevation. Through the inter- 
sections thus obtained trace a line, as indicated by G H. Then G II will be the miter line in elevation. At 
right angles to the gable lay off a stretchout of the profile at any convenient place, as shown by P R, through 
the points in which draw the usual measuring lines. Place the t-square parallel to the stretchout line, or, what 
is the same, at right angles to the lines of the gable, and, bringing it successively against the points in the miter 
line G H, cut the corresponding measuring lines. Since the surface against which the two moldings miter is 
that of a sphere, the pattern representing the space between the points 1 and 2 of the profile, and also between 
7 and 8 of the profile, will necessarily be an arc of a circle. Therefore in the pattern the line running from S 
to U, and also the line from V to T, must be struck from centers which are to be found. By inspection of the 
elevation it will be seen that the space S U is equal to that of D G struck from the center E. Set the dividers, 



168 



Pattern Problems. 



therefore, to E D or E G of the elevation, and from S and U respectively as centers, strike arcs, -which will he 
found to intersect at N". Then N is the center by which to describe the arc S U. By further inspection it will 

be seen that the hues corresponding to 7 and 8 of the stretch- 
out meet the profile of the ball at M. Continue this line 
indefinitely in the direction of K. From E, at right angles to 
it, draw the line E K. Then KMisa radius of the arc to be 
described from Y to T. Set the dividers to K M, and from 
V and T respectively as centers, strike arcs which will intersect 
in the point 0. From O, with the same radius, describe the 
arc Y T. Trace a line through the points from TJ to Y. 
Then SUYT is the pattern for the gable molding to fit 
against the ball, as shown. 

556. The inter Between the Moldings of Adjacent Gables 
iipon a Square Shaft, the Gables being of Different Pitches. — 
The problem presented in Figs. 433 and 434 is one occasionally 
arising in what is known as pinnacle work among cornice mak- 
ers. The figures represent the side and end elevations of a pin- 
nacle which is rectangular, but not square. Each of its faces is 
finished with a gable, which miter against each other at the 
eaves, and which are of the same hight iu the line of their 
ridges, as indicated by L M and L 1 M 1 . Whatever profile is 
given to the molding in one face of such a structure, the profile 
of the gable in the adjacent face will require some modification 
in order to form a miter. Let A be the profile of the molding 



Fig. 433.— Elevation of Side. 
The Miter Between the Moldings of Adjacent Gables 
Upon a Square Shaft, the Gables being of Dif- 
ferent Pitches. 

in the side. For the miter line in elevation and 
the pattern we proceed as follows : Draw a 
duplicate of A, placing it in a vertical position 
directly below or above the point at which 
the two moldings are to meet, as shown by A'. 
Divide both of these profiles into the same num- 
ber of parts, as indicated by the small figures, and 
through these points draw lines intersecting in the 
points from H to K. Then a line traced through 
these intersections, as shown by H K, will be the 
miter line in elevation. At right angles to the 
lines of the molding, as shown in elevation, lay 
off a stretchout of the profile A, as shown by 
B C, through the points in which draw the usual 
measuring lines. Place the T-square at right 
angles to the lines of the molding, or, what is the 
same, parallel to the stretchout line, and, bring- 
ing it against the several points in the miter line 
H K, cut corresponding measuring lines. Then 
a line traced through these points, as shown by D E, will be the shape of the cut at the foot of the side gable 
to miter against the adjacent gable. For the pattern of the piece to miter against the one just obtained, and 
belonging to the adjacent end, transfer the profile H K, reversing it, as shown by K 1 H l in Fig. 434, or, in lieu 





Fig. 434.— Elevation of End. 

The 3Iiter Between the Moldings of Adjacent Gables Upon a Square 

Shaft, the Gables being of Different Pitches. 






Pattern Problems. 



169 



of this, repeat the operation by which II K was obtained. At right angles to the line of the raking cornice in 
the end elevation, draw a duplicate of the normal profile, as shown by A 3 , which divide into the same number 
of equal parts as in the other case, and through the points carry lines across the cornice, as shown, intersecting 




c i a A ' 

Fig. 435.— Pattern for the Moldings and Poof Pieces in the Gables of an Octagon Pinnacle. 
these lines by lines drawn parallel to the lines of the cornice through the points in K 1 H 1 . Then a line traced 
through these points of intersection will form the modified profile, as shown by W X. For the pattern we pro- 
ceed as follows : At right angles to the lines of the rating cornice lay off a stretchout of the profile W X, as 
shown by P R, 'through the points in which draw measuring lines in the usual manner. With the T-square at 



170 



Pattern Problems. 



right angles to the lines of the raking cornice, or, what is the same, parallel to the stretchout line P E, bringing 
it successively against the points in K' II 1 , cut corresponding measuring lines. Then a line traced through these 
points of intersection, as shown from S to T, will be the pattern for the foot of the gable cornice on the end 
elevation. 

557. Pattern for the Holdings and Poof Pieces in the Gables of an Octagon Pinnacle. — Fig. 435 shows 

a partial elevation and a portion of the plan 
of an octagon pinnacle having equal gables 
on all sides. The first step in developing 
the patterns is to obtain a miter line at the 
foot of the gable, as shown by L. To do 
this proceed as follows : Draw the profile 
K, as shown, placing it so that it shall cor- 
respond in all its parts with the lines of the 
molding in elevation. Number spaces in it 
in the usual manner, as shown, and through 
the points draw lines parallel to the lines of 
the gable toward L, as shown. In the plan 
place a duplicate stay or profile, K, so drawn 
that its parts shall in all respects correspond 
to the position of those of the profile in the 
elevation. Divide it into the same number 
of spaces, and through the points in it draw 
lines parallel to the lines of the plan, cut- 
ting the line D F, representing the plan of 
the miter. From the points in D F thus 
obtained carry lines vertically, intersecting 
corresponding lines drawn through the pro- 
file in the elevation. A line traced through 
the several points of intersection, as shown 
by L, will be the line of miter in elevation 
between the moldings of the adjacent gables. 
The center line N forms the miter line 
for the top of the gable. For the pattern 
proceed as follows : Upon any line, as E E, 
drawn at right angles to the lines of the 
gable, lay off a stretchout of the profile, as 
shown by the small figures. Through the 
points of the stretchout draw the usual 
measuring lines. Place the J-square at 
right angles to the lines of the gable, and, 
bringing the blade successively against the 
points in the two miter lines above described, 
cut the corresponding measuring lines, as 
shown. Lines traced through the points of 
intersection thus obtained will give the pat- 
tern of the molding. The roof piece may 
be added by setting off A 1 B 1 at right 
angles to A 1 C, eqiml in length to A B of 
the side view. In like manner set off D 1 C 1 
equal to C D of the side view. Then draw 
F 1 D 1 B 1 , thus completing the pattern. 




Fig. 436.— Quarter Plan and Elevation of Wide Side. 

The Miter Between the Moldings of Adjacent Gables Upon an Octagon 

Shaft, the Gables being of Different Pitches. 



558. The Miter Between the Moldings of Adjacent Gables Vj:>on an Octagon Shaft, the Gables being of 
Different Pitches. — The problem illustrated in Figs. 430 and 437 resembles that presented in Sfction 556, save 



Pattern Problems. 



Ill 



that tlie angle is not a right angle. The elevations represent an octagon pinnacle of unequal sides, and the prob- 
lem is to cut the miter at the eaves occurring between adjacent gables of the same bight, but of different widths. 
Let A 1 B 1 F 1 O G 1 D 1 of Fig. 436 be a correct elevation, and A * 

ABCG be a quarter plan of the structure. In that portion 
of the plan corresponding to the part of the elevation shown 
to the front, draw the normal profile E, placing its vertical 
side parallel to the lines of the plan. Divide it into any con- 
venient number of spaces, and through these points draw lines 
parallel to the lines of the plan, cutting the miter line C O 1 , as 
shown. In like manuer place a duplicate of the normal pro- 
file, as shown by E' in the elevation. Divide it into the same 
number of equal parts, aud through the points draw lines par- 
allel to the lines of the raking cornice, which produce in the 
direction of 1ST O indefinitely. Bring the T-square against the 
points in C 0', and with it erect vertical lines, cutting the lines 
drawn through E 1 , as shown from N to 0. Then a line, N O, 
traced through these points of intersection will be the miter 
line in elevation. For the pattern of the foot of the gable 
shown in elevation proceed as follows : At right angles to the 
lines of the gable cornice lay off a stretchout of the profile E', 
as shown by H. K, through the points in which draw the usual 

measuring lines. Plac- 
ing the T-square at right 
angles to the lines of the 
cornice, or, what is the 
same, parallel to the 
stretchout line, - and 





bringing it against the' 
several points in N O, 
cut corresponding meas- 



Fig. 437.— Elevation of Narrow Side. 
The Miter Between the Moldings of Adjacent Gables 
Upon an Octagon Shaft, the Gables being of 
Different Pitches. 

uring lines. Then a line traced through the points of intersection thus 
obtained, as shown from L to M, will be the pattern of the foot of the 
galile shown in elevation. For the modified profile and the pattern of 
the gable piece forming the narrow side, proceed as follows : Transfer 
the miter line N O, reversing it, to the foot of the gable of the nar- 
row side, as shown by K P in Fig. 437, and through the points carry 
lines parallel to the lines of the gable cornice indefinitely, as shown. 
Draw a duplicate of the normal profile at any convenient point outside 
of the gable cornice, as shown by E 2 , placing its vertical side at right 
angles to the lines of the cornice. Divide E 2 into the same number of 
parts as used in the other profiles, and through the points draw lines at 
right angles to the lines of the cornice, intersecting the lines drawn from 
PE. Through these points trace a line, as indicated by E a , which will be 
the modified profile. Take the stretchout of E 3 and lay it off on any 
straight line drawn at right angles to the lines of the cornice, as S T, and 
through the points in it draw the usual measuring lines. Place the 
T-square at right angles to the lines of the gable cornice, and, bringing 
it against the points in P E, cut the measuring lines, as indicated by the 
The Pattern of a Square Spire Mitering dotted lines. Then a line traced through these points of intersection, as 
Upon Four Gables. shown by IT T, will be the pattern of the foot of the side gable. 

559. The Pattern of a Square Spire Mitering Upon Four Gables.— In Fig. 438, let B F IT C be the ele- 
vation of a square spire which is required to miter over four equal gables in a pinnacle, the plan of which is 
also square. Produce D B and E C until they meet in A, which will be the apex of the pyramid of which the 



172 



Pattern Problems. 



spire is a section. Draw the axis A G, and at right angles to it, opposite the lowest point of contact between 
the spire and the gable, as F, draw F G. Then F G will represent the half width of one of the sides of the 
pyramid at the base, and A F will represent the leDgth of a side through the center. From any convenient 
point, as A 1 in Fig. 439, draw A' F 1 , in length equal to A F. From F' set off, perpendicular to A 1 F", on each 
side a space equal to F G of the elevation, as shown by F 1 G 1 and F 1 G\ From G 1 and G ! draw lines to A 1 , 
as shown. From A 1 as center, and with A 1 G 2 as radius, describe an arc, as shown by G 2 0, in length equal to 
three spaces of the extent of G 1 G 2 , as shown by G 2 g, g cf and <f 0. Draw g' A", g A 1 and A 1 . Make 
A 1 E 1 equal to A B of the elevation, and through B 1 draw a perpendicular to A' F 1 , as shown. Draw lines 
corresponding to it through the other sections of the pattern. Make A 1 D 1 equal to A D, and draw D' G 1 and 

D l G 2 . Set the compasses to G 2 D 1 , and from G 5 
and g as centei-s, describe arcs intersecting at d. 
Draw d g and d G 2 , as shown. Repeat the same 
operation in the other sections of the pattern, 
thus completing the required shape. 

560. The Pattern of a Conical Spire Miter- 
ing Zfpon Pour Gables. — Let D B in Fig. 440 
be the elevation of a pinnacle having four equal 
gables, down upon which a conical spire is re- 
quired to be mitered, as shown. Produce the sides 
of the spire until they meet in the apex D. Also 
continue the side E F downward to any conveni- 
ent point below the junction between the spire 
and the gables, as shown by H, which point is to 
be considered the base of the spire. Let B 2 K L M 
be the plan of the pinnacle. The diagonal lines 
B 2 L and M K represent the angles between the 
gables, while B S and T TJ represent the ridges 
of the gables over which the spire is to be fitted. 
Through the point H in the elevation draw a line 
to the center of the cone, and at right angles to 
the axis, as shown by PI C. This will represent 
the half diameter or radius of the spire at the 
base. "With radius C H, and from center A' of 
the plan, describe a circle, as shown, which will 
represent the spire in plan. At any convenient 
distance from the elevation, and to one side, con- 
struct a diagonal section corresponding to the 
line B 2 A 2 in the plan, as shown. Draw D 1 A 1 
corresponding to the axis of the spire prolonged. 
At right angles to it, opposite the point B, set off A 1 B 1 equal to A 2 B 2 of the plan. Opposite F of the elevation 
establish F 2 in the line D 1 A 1 , and draw F 2 B 1 . Draw C 1 H 2 equal to and opposite C H in elevation. Draw IT D 1 , 
cutting B' F 2 in G 2 . From G 2 , at right angles to the axis, carry a line across the elevation, thus establishing 
the point G in the side of the spire. To describe the pattern, draw any straight line, as D 2 IF, Fig. 441, in 
length equal to the side D IT of the spire, and in it set off points corresponding to the points marked on 
the side of the spire. Thus make D 2 E 1 equal to D E, D 2 F 1 equal to D F, and D 2 G 1 equal to D G of the 
elevation. From D 2 as center, with D 2 E 1 as radius, describe the arc E 1 I 1 indefinitely. From the same center, 
with D 2 IP as radius, describe the arc II 1 V indefinitely. Divide one-quarter of the plan of cone into any 
number of equal spaces, and set off corresponding spaces from H 1 on the arc, as shown. Through the last of 
these points, as shown by 9, draw a line to the center D 2 , which in the pattern will be the same as B A 2 of the 
plan. In like manner, through the middle point, draw a line, as shown by 5 D 2 , which will correspond to B 2 A 
of the plan. From D 2 , with D 2 F 1 as radius, describe the arc F 1 X, and from the same center, with D 2 G 1 as 
radius, describe the arc G' "W. Then the arc drawn from F 1 will represent points which meet the tops of the 
gables, and that drawn from G 1 will represent the points in the angles between. Then the lines drawn from g* 




Fig. 439. — Development of the Pattern. 
The Pattern of a Square Spire Mitering Upon Four Gables. 



Pattern Problems. 



173 



to F 1 and g to f will be the required cut. On the arc H 1 V step off additional spaces, corresponding to the 
stretchout of a quarter of the plan of the cone, as shown from II 1 to 9, making four in all, and also mark the 
middle point (5) in each. Draw the line Y D 2 , 
and also the corresponding intermediate lines. 
Complete the pattern by drawing the diagonal 
lines corresponding to g f and g F 1 already de- 
scribed, all as shown in the engraving. 

561. The Pattern of an Octagon Spire 
Mitering Upon Four Gables. — In Fig. 442, let 
BEZC be the elevation of an octagon spire, 
mitering down upon four gables occurring upon 
a square shaft. Continue the side lines until they 
intersect in the apex A. Draw the center line 
A H, from which set off the perpendicular H G-, 
which shows the half width of one of the sides 
at the point G. Continue the side BE in the 
direction of F. Draw P F at right angles to the 
axis A H of the spire, thus establishing the point 
F, which shows the length of the sides fitting 
into the angle between the gables of the shaft. 
Draw A 1 F 1 in Fig. 443 equal to A F of the ele- 
vation, and set off points on it corresponding to 
points in A F. Thus make A 1 B 1 equal to A B, 
A 1 D 1 equal to A D, and A 1 E" equal to A E of the 
elevation, etc. Through E' draw a perpendicular 
equal in length to the width of a side at the point 
E, or double to G H, as shown in the elevation, 
placing one-half on each side from E 1 , all as 





the gables. 



Fig. 440.— Elevation, Plan and Section. 
The Pattern of a Conical Spire Mitering Upon Four Oablet. 

shown by L K. From L and K draw lines to A 1 , 
as shown. From A 1 as center, with A 1 L as radius, 
describe an arc, as shown by L IT, indefinite in 
length. Set the dividers to the space L K, and step 
off spaces from L, as L T, Y X and X U, until as 
many sides are set off as are required in one piece 
— in this case four. Draw the lines A' Y, A 1 X 
and A" U. By inspection of the elevation it will 
be seen that one-half the sides will be notched at 
the bottom to fit over the gables, while the others 
will be pointed to reach down into the angle between 
From the point D", which, as will be seen by D in the elevation, corresponds to the top of the 



Fig. 441.— Development oi the Pattern. 
The Pattern of a Conical Spire Mitering Upon Four Gables. 



174 



Pattern, Problems. 



gable, draw lines to the points L and K, which gives the pattern for the notch in the first section. Set the 
dividers to L D' as radius, and from X and Y as centers, describe arcs intersecting at W. Draw W X and W Y. 
For the pattern of the point place one leg of the dividers at L, and, bringing the other to F 1 , describe an arc, as 
shown by F 1 M. With the same radius, from Y as center, describe a second arc intersecting the first at M. Draw 
A MY and M L. Using the same radius, and U and X as centers, establish 

the point Y. Draw Y X and Y U, thus completing the pattern. 

562. The Pattern of an Octagon Spire Mitering Upon Eight Gables. 
— Let A C I in Fig. 444 be the elevation of the spire, and M 1ST O P the 
plan. From the point G, which represents the lowest point of the angle 
between the gables, to H, which represents the highest point of the 
gables corresponding to T in the plan, draw the line G H, cutting C Q in 
the point D. Draw any lint, as A 1 W in Fig. 445, upon which to con- 
struct the pattern. Make A ' Z equal to A C of the elevation, and A 1 W 
equal to A D of the elevation. Through W draw the perpendicular 1 Y, 
as shown. From W set off W Y equal to E F of the elevation, and 
likewise set off W 1 of the same length. Draw A 1 Y and A 1 1. Set the 
dividers to A 1 1 as radius, and from A 1 as center, describe the arc 1 U 
indefinitely. Set the dividers to 1 Y, and step off as many spaces on the 
arc as it is desired to have sides in the pattern — in this case four — as 
shown. Draw the lines A 1 TJ, A 1 3 and A 1 2, which represent the lines 
of bend in the pattern. Draw Z Y and Z 1 in the first section of the 
pattern. Set the dividers to Z Y, and from 1 and 2 as centers, describe 
intersecting arcs, as shown by Z'. In like manner describe similar inter- 
secting arcs at the points Z 3 and Z 3 . Draw lines from these points to the 
points 1, 2, 3 and U, as shown, thus completing the pattern. 

563. The Pattern of a Conical Spire Mitering Upon Eight Gables. 
Let E II C X I in Fig. 446 be the elevation of a pinnacle having eight 

Fig. 442. — Elevation. ° x o o 

The Pattern of an Octagon Spire Mitering equal gables, upon which the conical spire E F P I is to be fitted. Pro- 
Upon Four Gables. duce the sides F E and P I until they meet in the point D, which is the 

apex of the spire. Produce the side E F, continuing it downward 
until it meets the line of the vertical side of the shaft in G, which 
point is to be considered as representing the base of the spire. Let 
A IF S K M X 1 T IT be the plan of the pinnacle, which for conveni- 
ence draw in line with the shaft. From G in the elevation obtain 
the point G 3 in the plan, as shown by the dotted line. With radius 
B G", from B as center, describe the circle, as shown, which will 
represent a plan of the spire on a line through its base correspond- 
ing to the point G in the elevation. Divide an eighth of this circle 
into any number of equal parts, as shown by 1, 2, 3, 4 and 5, which 
spaces are to be used in measuring off the arc describing the pattern 
further on. To one side of the elevation, construct a diagonal sec- 
tion on the line A B of the plan, as shown. From the point H in 
the elevation, which corresponds to A in the plan, draw a horizontal 
line, as shown by B 1 A 1 , in length equal to B A of the plan. From 
B' erect a perpendicular, upon which locate the point D 1 , corre- 
sponding in hight to D. Mark the point F', corresponding to F of 
the elevation, and draw A 1 F'. Set off C'G' equal to and opposite 
C G in the elevation. Draw D' G", intersecting A 1 F 1 in the point 
R 1 . Bring the T-square against the point R 1 , the blade being at 
right angles to the axis of the spire, and draw R' P 4 , cutting the line 
E G at R. The point R 4 is of use in drawing the elevation, showing the extreme point of intersection between 
the spire and gable, but is not essential in cutting the pattern. Draw any line, as D 3 G 3 , Fig. 447, upon which 
to lay off the several points in the side of the cone. Make D 3 E 3 equal to D E of the elevation, D 3 F 5 equal 





Fig. 443.— Development of the Pattern. 

The Pattern of an Octagon Spire Mitering Upon 

Four Gables. 



Pattern Problems. 



175 



to D F, D' B' equal to D E, and D 2 G 2 equal to D G. From D 2 as center, describe the arc E' P indefi- 
nitely. Also from the same center, and with D 2 F 2 , D 2 E 2 and D 2 G' respectively as radii, describe the arcs 

F 2 X, E 2 W and G 2 V indefinitely. On the arc G 5 Y step off spaces 

corresponding to one-eighth of the plan, as shown by 1, 2, 3, 1 and 5, 

as already described. Draw the line D 2 5, as shown, and also draw a 

line from T>" to the middle point in the stretchout of the eighth of the 

circle, as shown by D 2 3. By inspection of the elevation it will be 

seen that the arc F 2 X represents the line at which the gables cut into 

the cone, and that the arc E 2 ~W represents the line of points in the 

base of the cone to fit down between the gables. Therefore from F 2 

draw F 2 g, and from a point in D 2 5 corresponding to F 2 , as /, draw 

f g. Then D'fg F 2 will be one eighth of the required pattern. Set 

the dividers to 1 5 on the arc G 2 V, and step off seven additional 

spaces, as shown by 5, 5, 5, etc., V. Draw V D 2 . Also draw the 

dotted lines D 2 5. From center point in each space, as indicated by 

3, draw lines to D 2 , as shown. Then draw the lines f g and g f, thus 

completing the pattern. 

561. The Gore Piece in a Holding Forming the Transition from 

an Octagon to a Square. — In Fig. 118, let A B C D be a half plan of 

the figure at the outside, and E F G II K L be a half of the inner 

plan. M E U P X is an elevation of the article, O E being the 

normal profile. F B G in the plan represents the transition piece by 

which the figure is changed from a square to an octagon, the pattern 

for which is desired. Draw the elevation and plan in such relative 

position that the profile O E shall fall directly over the miter line B F. 

Divide O E into any convenient number of 
spaces, and through these points drop lines 
cutting B F, all as indicated by the dotted 
lines. From the points in B F carry lines 
across the transition piece, parallel to F G, 
as shown, cutting the miter line G B, and 
from G B prolong them indefinitely. At 
right angles to the 
lines drawn through 
the transition piece 
just described, con- 
struct a duplicate of 
the normal profile, as 
shown by X T, which 
divide into the same 
number of parts as 
the original profile 
O E, and from these 
points, at right angles 

to the lines drawn across the transition piece, produce lines 
cutting lines of corresponding numbers. A line traced through 
the points of intersection thus formed, as shown by O 1 E', will 
be the profile of the transition piece. At right angles to F G 
and opposite the point B, draw V B', upon which lay off a 
stretchout of the profile O 1 E 1 . Through the points in B 1 V 

draw the usual measuring lines. Place the T-square parallel to this stretchout line, and, bringing it against the 

several points in the miter lines F B and G B, cut corresponding measuring lines, as shown. Then the lines 

traced through these points of intersection, as shown by F 1 B' and G' B 1 , will be the required pattern. 





Fig. 444.— Elevation and Plan. 

The Pattern of an Octagon Spire Mitering 

Upon Eight Gables. 



Fig. 445.— Development of the Pattern. 

The Pattern of an Octagon Spire ilitering Upon 

Eight Gables. 



176 



Pattern Problems. 



565. 
represent 



A Gore Piece Forming the Transition from an Octagon to a Square. — In Fig. 449, let F F F F 
the square plan of a base, and A A A A a portion of the plan of an octagon shaft which is to be 

fitted to it. Let DPCbe the elevation of the gore 
piece which is required to form the transition be- 
tween the two shapes. The rule which follows 
may be used for whatsoever profile the gore piece is 
required to be — in the present case an ogee. Since 
the outline of the gore piece, as shown by C D in 
elevation, is not a profile of the molding which 
forms the transition, it will be necessary to con- 
struct the profile of it before commencing the pat- 
tern. To do this proceed as follows : Divide the 
profile of the gore piece C D, as it appears in the ele- 
vation, in the usual manner, into any convenient 
number of spaces, as shown by 1, 2, 3, 4, etc. From 
the points thus obtained drop lines down upon one 
side of the plan A F, which is to be placed so that 
corresponding parts of plan and elevation shall 
aerree. Continue the lines from the side A F across 
the corner, as shown, all parallel to A A of the octa- 
gon. Bisect the angle A F A by the line E F, and 
upon it number the lines drawn across the corner 
to correspond with the numbers of the points in 
the elevation, from which they were derived. Draw 
G H parallel to P D, and at convenient distance 
from it. Cut G H by lines drawn at right angles 
to it from the points in the profile C D, as shown 
by the connecting dotted lines. G H then may 

V w 




K 

Fig. 446.— Elevation and Plan. 
The Pattern of a Conical Spire Mit-ering Upon Eight Gables. 

be considered to represent the point F in the plan, or 11 of 

the numbers on the line E F. From G H, on each of the 

several lines drawn through it, lay off a distance equal to 

the space from 11 on E F to the corresponding number on 

the same line. Thus lay off 11 1 from G H equal to 11 1 

on E K, and 11 2 equal to 11 2 of E K, and so on for each 

of the lines through G H. Then a line traced through 

these points, as shown by I H, will be the profile of the 

gore piece, or the shape of its section when cut by the line 

E F. Prolong E F, as shown by K L, and lay off on the 

latter a stretchout of the profile I H, the spaces of which 

must be taken from point to point as they occur, so as to 

have points in the stretchout corresponding to the points on the miter lines A F, A F, previously derived from 

D. Through the points thus obtained draw the usual measuring lines, as shown. Place the T-square at right 




Fig. 447.— Development of the Pattern. 
T'tie Pattern of a Conical Spire ifitering Upon Eight Gables. 



Pattern Problems. 



177 



angles to the measuring lines, or, what is the same, parallel to E K, and, bringing it against the points in A F 



and F A, cut the corresponding lines drawn through the stretchout. Lines traced 
shown, will constitute the pattern. 

566. The Blank for a Curved Holding.— Tigs. 450 and 451 are intro- 
duced at this place in order to show the principle upon which blanks for 
curved moldings are struck. In Fig. 450, A C E D represents the elevation, 
for example, of a wash basin, in which the sides are made bulging, as shown 
by the curved line from E to C. If the sides were straight, as shown by 
the straight line from E to C, the pattern would be easily described ; it 
would be simply the envelope of a section of a right cone. The patterns 
for curved moldings are cut upon exactly the same principle. The blanks 
are described in the same manner as though the article was to be formed up 
of a straight flare and not molded at all, save that additional width is 



through 



these points, as 




Pattern 



Fig. a 




Fig. 448. — The Gore Piece in a Mold- 
ing Forming the Transition from an 
Octagon to a Square. 



Fig. 449. — A Gore Piece Forming the Transition from an 
Octagon to a Square. 

given to it to compensate for the metal which is taken up by the 
form. Therefore, to describe the pattern of the blank from which 
to make a curved molding corresponding to the elevation A C E D, 
proceed in the same manner as though the side E C were to be straight. 
Through the center of the article draw the line B F indefinitely, and 
draw a line through the points C and E of one of the sides, which 
produce until it meets B F in the point F. Then F E will be the 

l J- > Fig. 450.— Obtaining the Sweep for a Simple Cove. 

radius of the inside of the pattern. The radius of the outside is to The Bfa; , fc for a Cuned Molding _ 

be obtained by increasing F C an amount equal to the excess of the 

curved line E C over the straight line E C, as shown by the distance C S. Then F S is the radius of the out- 
side of the pattern. In Fig. 451 the same operation is shown, applied to an ogee profile. The center line is 




178 



Pattern Problems. 



drawn. A line through the points of the profile is produced ttntil it cuts the center line in the point F. 

Then F is the center by which the arcs containing the pattern are to he struck. The distance between the two 

points E and C is extended to correspond to the stretchout of the profile. 
In cutting blanks for any metal molding, there must necessarily be some 
discretion exercised by the mechanic. Some sheets of iron will form 
more readily than others ; in some there is more stretch than in others, 
while the thickness has much to do with the operation of imparting the 
form to the blank. In certain kinds of metal it is found necessary to 
make arbitrary allowance, more or less in amount, to overcome difficulties 
peculiar to the material in hand. In the demonstration above given 
we have simply indicated the principles ; allowances are to be made as 
circumstances require. 

567. A Plain Window Cap and its Several Patterns. — One of the 
simplest articles to be made belonging to cornice work, and also one of 
the commonest for which patterns are demanded, is a plain window cap, 
yet in its various features it combines several of the most important prin- 
ciples in pattern cutting. In Fig. 452 we show the elevation and sec- 
tion of a very plain semicircular window cap, with corbels and keystone, 
of a style in quite general use. By inspection of the engravings it will 
be seen that a considerable portion of the patterns may be derived directly 
from these two views without any further drawing. The frame strip B 
and the roof A, for width are measured upon the section, and for length 
upon the lines in the elevation corresponding to them. The two flat 
face strips C and D are taken directly from the elevation. Set the divid- 
ers to the radii employed in striking them in the drawing, and lay off 
the sheet of iron. The face of the keystone must be transferred and shown in the 
This is a simple operation and may be performed as follows : 




Fig. 4si.— Obtaining the Sweep for an Ogee. 
The Blank for a Curved Molding. 



Face 
of Kev stone 



corresponding shapes upon 
flat, as indicated in the diagram at the right 
Upon any straight line, as P B, which shall 
be the center of the true face, lay off the 
length of the face taken from the section. 
Through the points thus obtained lay off the 
width of the face at top and bottom, as taken 
from the elevation. Then draw the con- 
necting lines. The patterns for the diamond 
ornament on the keystone are developed by 
means of the side elevation or section and 
two cross sections, taken through points cor- 
responding with the junction of the miter 
lines. This operation is clearly shown by 
the diagram entitled "Ornament on Key- 
stone," and needs no further explanation. 
The sides of the corbel are pricked direct 
from the sectional view. One side extends 
back on to the frame, as indicated by the 
dotted lines in the section, while the other 
extends back only to the face of the wall. 
The face of the corbel is set off upon the 
stretchout by measurement from the profile, 
as shown by the diagram entitled "Face of 
Corbel." The method of developing the 

pattern for the curve molding is also dearly shown in the diagram bearing that name. Through the center M, 
by which the elevation lines of the molding were struck, draw a horizontal line, M F, at convenience. Upon 
any point in it outside of the elevation, as li, erect a perpendicular, II I, which in length make equal to the 




Pattern for Miter of 
Mo.lding against Corbel 



Ornament 
on Keystone 



Fig. 452. — A Plain Window Cap and its Several Patterns. 



Pattern Problems. 



179 



radius by which the inner line of the molding was drawn. Upon I construct a profile of the curved molding, 
as shown by G T, placing it as it would appear if the cap were cut through the center line which II I may be 
supposed to represent. Through the extreme points in the profile of the curve draw a line, G T, as shown, 
which prolong until it cuts the 
line M F in the point F. Then 
F T will be the radius of the 
inner edge of the blank for 
the curved molding. From T, 
on the line T G, lay off a 
stretchout of the profile, thus 
obtaining the point G. Then 
F G will be the radius of the 
outer line of the pattern. The 
straight portion K is added 
by drawing a line across the 
pattern through the center by 
which it is struck, as shown 
by the dotted line TJ F, the 
lines of the straight part be- 
ing produced to the required 
length at right angles to it. 
In the construction of window 
caps it is usual to trim the 
lower end of the cap C D 
after the several parts are 
joined, in order to fit it against 
the sloping top of the corbel. 
For this purpose a miter pat- 
tern is shown at L correspond- 
ing to the bottom of the win- 
dow cap as it is to be cut. 
This is used simply as a gauge by which to scribe the line to which the cap is to be trimmed, in order to fit the 
corbel. Many workmen cut by eye without this pattern and with good results. Lay off a duplicate of the pro- 
file of the cap, as shown at L, which divide into any convenient number of equal parts. Draw a miter line, as 

shown by * y, corresponding to the pitch of the top of the corbel. Drop 
lines from the point of the profile against the miter line, and then with the 
T-square placed at right angles to the lines drawn from the profile, cut cor- 
responding measuring lines in a stretchout previously laid off, as shown. 
A line traced through these points in the measuring lines will be a pat- 
tern of the shape to which the end C D must be cut to fit the top of the 
corbel. The molding around the top of the keystone consists of three pieces, 
joined by simple square miters. Divide the profile of the cap molding into 
any convenient number of parts. At right angles to the lines of the mold- 
ing lay off a stretchout, and through the points in the stretchout draw meas- 
uring lines in the usual manner. With the T-square placed at right angles 
to these measuring lines, and brought successively against the jioints in the 
profile, cut the measuring lines. A line traced through these points will 
give the shape of the required patterns. By extending the measuring lines 
through the stretchout across the space over the elevation and section, and 




Fig. 453.— Elevation of the Window Cap. 
The Patterns for Simple Curved Moldings in a Window Cap. 




-Laying out the Blank for Center 
Piece. 



The Patterns for Simple Curved Mold- 
wiQS in, a Windoiv Cap. 

by dividing each of the several profiles of the cap molding shown in the elevation and section and dropping 
points from all, the two patterns required are produced complete, all as indicated in the engraving. Flanges 
and joints are to be added at the discretion of the mechanic. 

56S. The Patterns for Simple Curved Moldings in a Window Cap.— In Fig. 453 we show the elevation 



180 



Pattern Problems. 



of a window cap, 
opposite sweeps. 



/^ 


- 


/--- 


2 


iL 


3 -" 




Fig. 455.— Laying out the Blank for Side Piece. 

The Patterns for Simple Curved Moldings in 

a Window Cap. 



in the construction of which two curved moldings are required of the same profile, hut of 
The only features peculiar to this cap are the patterns for these curved moldings, which, 

therefore, are the only parts we shall describe. The patterns for 
the returns, the corbels and the face portion of the cap are ob- 
tained in the same manner as corresponding portions of other 
caps elsewhere described. The profiles S and R have fillets, and 
are to be constructed with riveting edges, the whole of which it 
is possible to raise in one piece. The method of developing the 
pattern for the blank is the same for both curves. The two 
pieces will raise to the form by the same dies or rolls, it being 
necessary only to reverse them in the machine. For the patterns 
proceed as follows : From G, the center by which the curve in 
the middle part of the cap is struck, draw A D at right angles to 
the center line of the cap, as shown. At convenient distance 
from the center line, and parallel to it through D, draw H K. 
Draw the profile S, placing it on H K in such position that its 
several members shall be as far removed from the point D as the 
corresponding members of the molding in elevation are removed 
from the center G. The dotted lines running from the profile to the elevation show the correspondence of the 
parts. By this arrangement it will be seen that the profile S, in connection with the line H K, represents a sec- 
tion of the structure about 
to be constructed, of which 
A D is the center line. The 
principle to be employed 
in striking the pattern is 
simply that which would be 
used in obtaining the envel- 
ope of the frustum of a 
cone. The general average 
of the profile is to be taken 
in establishing the section 
of the cone, or, in other 
words, a line is passed 
through its extreme points. 
Draw a line through the 
profile in this manner and 
prolong it until it inter- 
sects A D in the point A, 
all as shown by C A. Then 
A is the apex of the cone, 
of which the profile S may 
be considered a section. 
Divide the profile S, as 
in ordinary practice for 
stretchouts, into any num- 
ber of spaces, all as shown 
by the small figures. Trans- 
fer the stretchout of the 
profile S on to the line A C, 
commencing at the point 1, 

as shown, letting the extra FiS- 456.-Elevation and Section of Cap. 

width extend in the direc- ■^ le Patterns for Elliptical Curved Moldings in a Window Cap. 

tion of C. From any convenient center, as A 1 in Fig. 454, with radius A 1 C 1 , describe the pattern, making the 



ti'on 



1-^- 




Pattern Problems. 



181 



length of the arc equal to the length of the corresponding arc in the elevation, all as shown by the spaces and 
numbers. For the pattern of the curved molding forming the end portion of the cap proceed in the same o-en- 
eral manner. Draw a profile, R, as shown, placing it against the line L M drawn through the center F, by 
which the curve in the elevation is struck. Through 
F draw the perpendicular F B indefinitely. Through 
the average of the profile R, as before explained, 
draw the line E B, cutting F B in the point B, as 
shown. Lay off the stretchout of the profile upon 
this line, commencing at the point 1, in the same 
manner as explained in the previous operation. 
From any convenient point, as B 1 in Fig. 455, with 
radius B E, describe the pattern, as shown from E 1 
to E\ which in length make equal to the arc repre- 
senting the same curve in the elevation, all as shown 
by the measurements indicated by the small figures. 
The straight portion forming the end of this mold- 
ing, as shown in the elevation, is added by drawing, 
at right angles to the line E 2 B 1 , a continuation of 
the lines of the molding of the required length, as 

shown in the pattern. Upon this end of the pattern a square miter is to be 
cut by the ordinary rule for such purposes, to join to the return at the end 
of the cap. 

569. The Patterns for Elliptical Curved Moldings in a Window Cap. — 
In Fig. 456 we show the elevation and vertical section of a window cap elliptical 
in shape, the face of which is molded. The patterns for the corbels, roof and 
frame strips have no peculiar features about them, and therefore will not be 
described in this connection. For the pattern of the elliptical curved molding 
we proceed as follows : In drawing the elevation certain centers were employed, 
or if the elevation was struck after the manner of drawing an ellipse — with 
string and pencil or by trammel — then the centers of approximate arcs must be 
obtained, the number of which, in either case, depends upon the accuracy with 
which the elliptical curve has been drawn. Having in this way determined 
the centers B, D and F, by which the respective sections of the elevation are or 
may be struck, use them in obtaining patterns as follows : Through the center 
F, from which the arc forming the middle part of the cap is drawn, and at 
right angles to the center line of the cap G H, draw the line I K indefinitely. 
Through the average of the profile, as indicated, producing the line until 
it meets I K, draw S R. Divide the profile in the usual manner and lay off 
the stretchout, as indicated by the small figures. Then R S is the radius of the 
pattern of the middle section of the cap. From K, through the section, erect 
K P, as a common basis of measurement by which to obtain the radii of the other 
portions. "With the dividers, measuring down from the profile, lay off on P K 
distances equal to the length of the radius A B, as shown by the point 0, and 
C D, as shown by the point M. Through these points O and M, at right angles 
to P K, draw lines cutting S R in the points T and IT. Then IT S is the radius 
for the pattern of the second section of the curve, and T S the radius of the pat- 
tern for the third section of the curve. In order to obtain the correct length of 
the pattern, not only as regards the whole piece, but also as regards the length of 
each arc constituting the curve, step off the length of the curved molding with the dividers, as shown in the 
elevation, numbering the spaces as indicated. As a matter both of convenience and accuracy, the spaces used 
in measuring the arcs are greater in the one of larger radius and are diminished in those of shorter radii, as 
will be noticed by examination of the diagram. To lay off the pattern after the radii are obtained as above 
described, proceed as follows : Draw any straight line, as G 1 II 1 in Fig. 457, from any point in which, as F 1 , 




Fig. 457.— Laying off the Pattern. 

The Patterns for Elliptical Curved 

Moldings in a Window Cap. 



182 



Pattern Problems. 




with radius equal to E. S, as shown by F 1 E 1 , describe an arc, as shown by E 1 G 1 ; and likewise, from the same 
center describe other arcs corresponding to other points in the stretchout of the profile. Make the length of 
the arc E 1 G 1 equal to the length of the corresponding arc in the elevation. From E 1 to the center F 1 , by which 
this arc was struck, draw E 1 F 1 . Set the dividers to the distance TJ S as radius, with which, measuring from E 1 
along the line E 1 F 1 , establish D 1 as center, from which describe arcs corresponding to the points in the profile, 

as shown from E 1 to C, which in turn make equal to the length of 
the corresponding arc in the elevation, all as shown by the small 
figures. From C draw the lino C 1 D l to the center by which this 
arc was struck. Set the dividers to the distance F S in the elevation, 
and measure from C along the line C 1 D\ Establish the point B 1 
for center, from which strike arcs corresponding to those already 
described in the other section of the pattern. Make the length 
equal to the length of the corresponding sections in the elevation, 
and draw the line A 1 B 1 . Then A 1 C 1 E 1 G 1 is the half pattern 
corresponding to A C E G of the elevation. 

570. Patterns of the Pace and Side of a Plain Tapering Key- 
stone. —Let A B D C in Fig. 458 be the elevation of the face of a 
keystone, and G E 1 P K of Fig. 459 a section of the same on its cen- 
ter line. For the true face and side, or, in other words, for the pattern 
of the face and side, proceed as follows . Through the center of the 
face draw E F, which prolong indefinitely. Through any convenient 
point in E F prolonged, as E 1 , and at right angles to it, draw A 1 B 1 , 
equal to A B of the elevation. Set off E 1 F 1 equal to E 2 F 2 of the 
sectional view, and through F 1 , at right angles to E 1 F 1 , draw C 1 D 1 , 
in length equal to C D, as indicated by the dotted lines. Connect 
A 1 C 1 and B 1 D 1 . Then A' B 1 C 1 D l will be the pattern for face of 
keystone. For the side we proceed as follows : Produce II K of Fig. 
459 indefinitely, as shown by L K', and at any convenient point erect the perpendicular 
L 1 E 3 , letting the point E 3 fall directly under E 2 of the side view, as shown by the 
dotted lines. Make L K 1 equal to B D of the elevation, and from K 1 erect the per- 
pendicular E? F 2 equal to K F 2 of the sectional view, as shown by the dotted lines. 
Connect E 2 and F 3 . Then E 3 G 1 B? F 3 will be the outline of the side of the keystone. 
Make L M equal to B M 1 of the elevation, and make L H' equal to B H 2 of the eleva- 
tion. Then with M H 1 as a basis of measurement draw N 1 as a duplicate of the profile 
N in the side view, thns completing the pattern of the side. 

571. Patterns for a Keystone with Sink in Pace.— In Fig. 460, E A B F repre- 
sents the face of a keystone, as for a window cap, fitting over a molding, as shown in 
profile by M 1ST O. In the face there is a sink, shown by G H D C, extending through 
the length of the keystone. L Iv B S represents the profile of the face of the key- 
stone, and K T represents the profile of the sink in the face. By the conditions as 
thus described it will be seen that the face of the keystone tapers, that its profile is 
irregular, that the profile of the sink in the face does not correspond to the profile of 
the face, and that the sink also tapers, being wider at the top than at the bottom. For 
the several patterns involved proceed as follows : Divide the profile of the face 
K B into any convenient number of spaces, and from the points thus obtained 
carry lines across the face of the keystone, as shown. Since K R represents the 
profile of the face, a stretchout taken from it is to be used by which to locate the 
measuring lines upon which to drop points from the face piece. At right angles to the 
keystone lay off a stretchout of K B, as shown by K 2 B', through which draw the usual measuring lines. 
Placing the T-square parallel to the stretchout line, and, bringing it successively against the points in the lines 
C D and B A bounding the face strip, cut the corresponding measuring lines. Then a line traced through these 
points, as shown by C 3 A 5 B 2 D 3 will be the pattern for this part. For the pattern of the sink piece, as shown 
in elevation by G D C H, the profile K T is to be used. The usual method would be to divide K T into equal 



Fig. 458.— Elevation. 

Patterns of the Face and Side of a Plain 

Tapering Keystone. 




Fig. 459. — Section. 
Patterns of the Face and 
Side of a Plain Taper- 
ing Keystone. 



Pattern Problems. 



183 



spaces, carrying lines across the face ; but since this would result in confusion, we have used the same points as 
established in K E, which are quite as convenient for use as the others mentioned, save that in laying off the 
stretchout each individual space must be measured by the dividers. At right angles to the line II I> of the key- 
stone lay off a stretchout of K T, as shown by K 1 T", through the points in which draw the usual measuring 
lines. Place the T-square at right angles to the lines across the face of the keystone, and, bringing it successively 
against the points in the lines G IT and C D, forming 
the sides of the sink, cut the corresponding measuring 
lines drawn through K 1 T 1 . Then lines traced through 
these points, as indicated by G' II 1 and C l D 1 , will 
form the pattern of the required sink piece. For the 
pattern of the strip forming the sides of the sink in 
the face of the keystone, at any convenient place in 
line with the side view of the bracket, lay off a space 
equal to the side strips, as shown in the face by C D. 
Transfer to that line the several points in C D, as 
determined by the lines crossing it drawn from the 
profile, all as indicated by C 2 D 2 . 
Through the points in C D 2 draw 
measuring lines in the usual manner. 
Place the T-square at right angles to 
these measuring lines, and, bringing it 
successively against the several points 
in the profiles K B and K T, cut 
the corresponding measuring lines, as 
shown. Then a line traced through 
these points, as indicated by K 3 B 2 
and K 3 T 2 , will be the pattern of the 
strip required. In this connection it is 
proper to remark that while using the 
same points in the profile Iv T as we 
use iu K E, although a matter of some 
inconvenience in describing the pat- 
tern of the sink strip, mention of 
which was made above, it would be 
still more inconvenient in describing 
the pattern last explained if the points 
of the two profiles were not derived 
from the same source. In other 
words, if the points in the profile 
K T were established arbitrarily and 
were entirely independent of those in 
profile K E, it would necessitate two sets of measuring lines drawn through the stretchout C 2 D 2 , resulting 
in great confusion. For the side of the keystone we proceed in the same manner as described in connec- 
tion with the sink strip just explained. Lay off A 1 B', in length equal to the side A B of the keystone, putting 
into A 1 B 1 all the points occurring in A B, through which draw measuring lines in the usual manner. Place 
the T-square at right angles to these measuring lines, and, bringing it successively against the points in the pro- 
file K E, also against points in the molding IS" and O, and likewise against L 31 and P S of the back, cut corre- 
sponding measuring lines, as shown. Then a line traced through these points of intersection, as shown by 
N 1 3P L 1 K 1 E 2 S' P 1 O 1 , will be the outline of the required pattern, with the exception of that part lying 
between X 1 and O 1 , which make a duplicate of 1ST 0. By examination of the points in A 1 B' and the lines 
drawn through the same, making comparison with the points in A B, it will be seen that in order to locate all 
the points in the profile of the molding 31 1ST P, two additional points, as shown by W and O 1 , have been 
marked in A 1 B 1 , corresponding to the points of intersection between the extreme lines of the molding itself 




Fig. 



H' r o' 

460. — Patterns for a Tapering Keystone with Sink in Face. 



184 



Pattern Problems 



molding. 



In practice it is frequently 



and the side A B, as shown in the elevation by the curved lines of the 
necessary, in operations of this character, to introduce extra points. 

572. The Patterns for a Palling Bracket.— La. Fig. 461 of the accompanying engravings, L P Q repre- 
sents the normal profile of a bracket, corresponding to which a raking bracket is to be constructed. K O P L 
represents the face view of the raking bracket as it is required to be. In the side view the dotted line U D 

V M> 




represents the profile of the sink in 
the face, which is shown in the front 
view by E F H G. In the side view 
a d c l> represents the shape of the 
panel in the side, of which ABDC 
in the face view shows the depth. 
For the several patterns required jsro- 
ceed as follows : For the top molding 
of the bracket the first stej) is to draw 
the face view as it would appear when 
constructed, thereby getting in eleva- 
tion miter lines by which to work. 
Divide the normal profile L ]ST into 
convenient spaces, and from the points 
thus obtained carry lines indefinitely 
parallel to the rake. Across the top 
of the face view of the bracket draw 
duplicates of the normal profile, plac- 
ing them in a vertical position directly 
above where the new sides are required 
to be, as shown by n I and h in. Di- 
vide these two profiles into the same 
number of parts employed in dividing 
" the normal profile, and from these 
points drop lines vertically, intersect- 
ing those drawn from L N. Then a 
line traced through these points of 
intersection, as shown by L 1 N* and K M, will be respectively the profile of the molding on the upper side and 
on the lower side of the bracket. At right angles to the line of the rake lay off a stretchout of the profile n I, 
or, in other words, the normal profile, as shown by L 1 IS. 1 , and through the points in it draw the usual measuring 
lines. With the blade of the T-square at right angles to the lines of the rake, and brought successively against 
the several points in the profile N 1 L 1 and R M, cut the measuring lines drawn through the stretchout. Then 
a line traced through the points of intersection thus obtained, as shown by 1/ W and K 1 M 2 , will be the shape 



Fig. 461.— Elevation and Shapes of the Principal Parts. 
The Patterns for a Raking Bracket. 



Pattern Problems. 



185 




Fig. 462.— Upper Return of Head. 

The Patterns for a Baking 
Bracket. 



of the ends of the molding forming the front of the bracket head. By observation it is evident that in 
forming this molding the normal profile is to be used as a stay, which is to be placed at right angles to the lines 
of the molding. For the return moldings, forming the sides of the bracket heads, a duplicate of the profile 
L 1 X 1 is transferred to any convenient place, as shown by L 3 X 1 in Fig. 4G2. By this a representation of the 
side of the head is drawn, making X 4 X 2 equal to X X of the side view of the bracket. Space the profile of 
the ends of this side view into any convenient number of parts, as shown by the small figures in L 3 X 1 and 
Q 1 X 2 . At right angles to the lines of the molding lay off a stretchout of these 
profiles, as shown by q x, and through the points in it draw the usual measuring- 
lines. "With the T-square at right angles to the lines in the molding, and brought 
successively against the points in the profiles L 3 X 4 and Q 1 X", cut the correspond- 
ing measuring lines. Then lines traced through these points of intersection, as 
shown by L 4 X s and Q 2 X 3 , will form the pattern. The pattern for the return 
molding of the head occurring on the lower side of the bracket is obtained in 
the same manner. A duplicate of the profile EM of the face view of the 
bracket is drawn at any convenient place, as shown by X 2 M 2 in Fig. 463. The 
proper length is given to the molding by measuring upon the side view of the 
bracket, and a duplicate profile is drawn at the opposite end. Space the profile 
X s M" into any convenient number of parts, as indicated by the small figures, 
and in like manner into the same number of parts divide the profile X 3 M 3 . At 
right angles to the lines of the molding lay off a stretchout of these profiles, as 
shown by X 1 M 1 , through which draw the usual measuring lines. With the blade of the T-square at right angles 
to the lines of the molding, and brought successively against the several points in the profiles X 2 M 2 and X 3 M 3 , 
cut the corresponding measuring lines. Then a line traced through these points of intersection, as shown by 
X 5 M s and X 4 M 4 , will constitute the pattern of the return molding, or the lower side of the bracket. For the 
patterns of the several pieces forming the face of the bracket, the profile, as shown in the side, is divided into 
any convenient number of spaces, and through the points thus obtained lines are drawn parallel to the lines of 
the rake, crossing the face of the bracket ; stretchouts are taken from the several profiles in the side view and 
laid out at right angles to the lines of the rake, through which the usual measuring lines are drawn. Points in 
the several pieces composing the face are then dropped upon these measuring lines, giving points of intersection 
through which lines are traced constituting the several patterns. For the strip BEGS, forming the face at 
the side of the sink, the profile U Z is subdivided, as indicated by the small figures, and lines from these points 
are carried across K E G S, as shown. At right angles to the lines of the rake a stretchout of the profile U Z 
is laid off, as shown by u" s% through the points in which the usual measuring lines are drawn. With the 
T-square placed at right angles to the lines of the rake, and brought successively against the points in the sides 
B S and E G, the corresponding measuring lines are cut. Then lines traced through these points of intersec- 
tion, as shown by B 1 S 1 and E 1 G 3 , form the pattern for that piece. For the 
piece forming the face of the bracket below the sink, as shown in the elevation 
by S B 1 Z', proceed in like manner. The profile Z B in the side view is 
divided into any convenient number of parts, and through the points lines are 
drawn, crossing the face as shown. A stretchout, as indicated by cV p, is laid off 
at right angles to the lines of the rake, through which the usual measuring lines 
are drawn. The T-square is then placed at right angles to the lines of the rake, 
and brought against the several points in the sides S O and Z 1 B 1 , by which the 
corresponding measuring lines are cut. In like manner it is brought against the 
points G and H, by which the shape of the part extending up to meet the sink is 
determined. Then lines traced through these several points of intersection, as 
shown by H 3 Z 3 B 3 O 1 S 2 G\ form the pattern for that part of the face of the bracket. The upper part of the 
face of the bracket, shown in the face view by X ! U 1 B ]tf, being a flat surface, as indicated in the side view 
X U, is obtained by pricking directly from the face view of the bracket. Xo development of it is necessary. 
To avoid confusion of lines, the sink piece E F H G is transferred to the right, as shown by E 1 F 1 H 1 G'. The 
profile of it, as indicated in the side view by II D, is divided into any convenient number of spaces, and through 
the points lines are drawn crossing it. The stretchout of this profile, as shown by u l d", is laid off at right 
angles to the lines of the rake, and through the points in it the usual measuring lines are drawn. The T-square 




Fig. 463.— Lower Return of Head. 

The Patterns for a Baking 

Bracket. 



186 



Pattern Problems. 



is then placed at right angles to the hues of the rake, and, being brought successively against the points in the 
sides h la and h 11 , the corresponding measuring lines are cut. Then lines traced through these points of 
intersection as shown by E" & F W, constitute the pattern of the bottom of the sink. Of the strips bound- 
ing the panel of the side m the bracket, the piece corresponding to I e in the side view is obtained by pricking 
directly from the face view of the bracket, A B D' C being the shape. For the other straight strip bounding 




Fig. 464. — A Raking Bracket in a Curved Pediment. 
this panel, as shown in the side view by a I, the length is laid off equal to a b, while the width is taken from 
the f aC e view, equal to the space indicated by A B. For the strip representing the irregular part proceed as 
follows : Divide the profile a d c into any convenient number of parts, from the points in which carry lines 
crossing the face view of the same part, as indicated by ABD'C. At right angles to the lines of the rake lay 
off a stretchout of the profile just named, as indicated by a' &, through the points in which draw the usual 



Pattern Problems. 



187 



measuring lines. 



Place the "T-square at right angles to the lines of rake, and, bringing it against the several 
points in the line A C and B D', cut the corresponding measuring lines drawn through the stretchout. Then 
lines traced through the several points of intersection thus formed, as indicated by A 1 C 2 and B' D 3 , will be the 
pattern of the curved strip forming part of the boundary of the panel in the side view of the bracket. For 
the side of the bracket, including the bottom of the panel last described and the strips forming the sides of the 
sink in the face of the bracket, we proceed as follows : Through the several points already established in 
the profile of the bracket, as shown by the side view, and in the profile of the sink and the shape of the panel, 
likewise shown in the side view, carry lines parallel to the rake, intersecting any vertical line, as X 1 P 2 . From 
the points thus obtained in the line X 1 P 2 , carry lines indefinitely horizontally, as indicated. Upon each of the 
lines so drawn lay off from the line X 1 P 2 a distance or distances equal to the distance or distances upon the 




Fig. 465. — The Principles upon ivhich the Plain Surfaces of a Mansard Finish are Developed. 

corresponding lines drawn across the normal side of the bracket. Through the points thus obtained trace lines, 
which will give the several shapes in the sides of the brackets corresponding to the shapes shown in the normal 
side of the bracket. It may be necessary to introduce in the several profiles of the normal bracket other points 
than those which have been used in developing the patterns described. Use as many points in the several pro- 
files in the normal side of the bracket as may be necessary to determine, the points in the side being constructed. 
Then X' 1ST 2 P 2 will be the pattern of the side of the bracket, and U 2 72 D 2 will be the pattern of the strip 
forming the sides of the sink shown in the face by E F II G, and V a 1 d 1 & will be the shape of the panel in 
the side of the bracket. 

573. A Raking Bracket in a Curved Pediment— Let E A C in Fig. 464 be the arch to which the bracket 
is to be fitted. C K is the center line of the pediment. Draw the normal profile of the bracket with its back 
against the center line, as shown by C D G. Divide the face of this profile into any convenient number of 
parts, as shown by the small figures, and from these points carry lines at right angles to the back of the bracket, 
cutting the lines C G, as shown. Thence carry lines around the arch from the center K, by which the 



188 



Pattern Problems. 



same is struck. Let E A B F be the face of the raked bracket as it will appear in elevation. Terminate the 
arcs corresponding to the points in C G, struck by the center K against the side A B. Draw lines through the 
points E A and F B, which produce until they intersect in the point X. From X draw lines through each 
point in A B, crossing the bracket, as shown, continuing them until they cut the side E F in the points a, b, c, 
d, etc. Draw a duplicate of the normal profile below and to one side of the face E A B F, as shown by S T H. 
Divide the line of the face S H into the same number of parts as used in the division of the face D G, and 
from these points carry lines upward parallel to the back T II indefinitely. Produce the line of back T H, ver- 
tically, upon which to construct the profile of the side E F. Place the T-square at right angles to the side E F, 
and, bringing it against the several points E, a, b, c, etc., in it, cut corresponding vertical lines drawn from the 
normal profile S H. Then a line traced through these points, as shown by L F, will be the profile of the lower 
side of the bracket. Still further produce the line H T, as shown by B 1 A 1 . Make B 1 A 1 equal to the upper 




Fig. 466. — Elevation and Development of Patterns. 
The Patterns of a Hip Molding upon a Right Angle in a Mansard Roof, Mitering Against the Planceer of the Deck Cornice. 

side of the bracket B A, as shown in the elevation, and set off in it points corresponding to the points in B A, 
through which draw lines at right angles to B 1 A 1 . Intersect these lines in turn by lines drawn from points in 
the profile S H, and through these points of intersection draw a line, as shown, from M to B 1 , Then M B 1 
represents the profile of the upper side of the bracket. For the face of the bracket proceed as follows : Lay 
off E' X 1 at any convenient place, in length equal to E H. From E 1 as center, with radius equal to 1 2 of the 
profile L F 1 , describe an arc, as indicated, and from X 1 as center, with radius X A, describe an arc intersecting 
the other in the point a\ Draw a 1 X 1 , and to this line, at each extremity, erect perpendiculars, as shown by 
a 1 a~ and X 1 3, in length equal to the spaces between the points 2 and 3 of the profile L F'. Draw a" 3. "With 
a~ as center, and with radius equal to 3 4 of the profile L F 1 , describe an arc, as shown. From 3, in the line 
X' Z, as center, with radius X b, describe an arc, intersecting it in the point b l . From 3, in the line X 1 Z, erect 
a perpendicular to*the line A 2 3, in length equal to the difference in the projection between the points 3 and 4 
of the profile C F' as measured upon the line S T, as indicated by 3 4. Draw the line 4 b\ Proceed in the 



o 



Pattern Problems. I39 

same manner from this base, obtaining the point c\ using 4 5 of the profile L F 1 and X c as radii for arcs inter- 
secting in the point C. For the space 4 5 take the difference in the projection between the points of corre- 
sponding numbers in the profile L F', as measured upon S T, setting it off each time perpendicular to the line 
from which it is drawn, continuing in this manner until all the points are used. Then a line traced through the 
points E 1 , a 1 , a", IS, e\ etc., to F 2 will be the shape of the edge of the face corresponding to the lower side of the 
bracket. On each of the lines corresponding to these several points, E', a 1 , a\ V, etc., set off a width ecpial to 
the width of the face EABF, measured on corresponding lines. Then a line traced through the points thus 
obtained, as shown by A 2 B 2 , will be the shape of the face corresponding to the upper line of the bracket. 

574. The Principles upon which the Plain Surf 'aces of a Mansard Finish are Developed. One of the 

first steps in developing the patterns for trimming the angles of a mansard roof is to obtain a representation of 
the true face of the roof. In other words, inasmuch as the roof slopes in two ways, the length of the hip is 
ather than is shown in the elevation, and this difference extends in a proportionate degree to the lines of the 
various parts forming the finish. The true face of a mansard may be obtained by" either of the following 
methods : In Fig. 465, let A E F C be the elevation of a mansard roof as ordinarily drawn, and let A CI be 
the profile or pitch drawn in line with the elevation. Set the dividers to the length A 1 G, and from A 1 as cen- 
ter, strike the arc CI (I 1 , letting CI 1 fall in a vertical line drawn from A 1 . From CI 1 draw a line parallel to the 
face of the elevation, as shown by G 1 C, and from the several points in the corner finish, as shown by C and K, 
drop lines vertically, cutting G 1 C 1 in the points C 1 and Iv 1 , as shown. From these points carry lines to corre- 
sponding points in the lipper line of the elevation, as shown by C A and K 1 h. Then 

AC'F'E represents the pattern of the surface shown by A C F E of the elevation. 
In cases where the whole bight of the roof cannot be put into the drawing for use, as 
above described, the same result may be accomplished in the following manner: Es- 
tablish any point, B, in the line of the hip, and from A, in a vertical line, set off A B 1 , 
equal to A B. From B 1 draw the horizontal line, as shown by B 1 B 3 , and from B drop 
a vertical line cutting this line, as shown, in the point B 3 . By inspection of the engrav- 
ing it will be seen that the point B 3 falls in the line A C 1 previously obtained, thus 
demonstrating that the latter method of obtaining the angle by which to proportion 
the several parts corresponds to the method first described, and therefore may be used 

, x . x J The Patterns of a Hip 

when more convenient. _ _ Molding Upm a BjgU 

575. The Patterns of a JI/'p Molding upon a Right Angle in a Mansard Poof, Angle in a Mansard 
Miter ing Against the Planceer of a Peck Cornice. — Let Z X Y V in Fig. 46G be the Roo f> Altering Against 
elevation of a deck cornice, against the planceer of which a hip molding, IT W Y T, the Planceer °f the De °* 

Cornice. 

miters. Let the angle of the roof be a right angle, as shown by the plan Q D A 1 , Fig. 

467. The first step in the development of the patterns will be to construct a diagonal elevation of the hip molding. 
Assume any point, A, in the elevation on a line drawn through the fascia of the profile, as shown by B A. 
Through A draw a horizontal line indefinitely, as shown by L A C. From B, the point in the line A B against 
the planceer, drop a vertical line, cutting the horizontal line drawn through A at the point C, all as shown by 
B C. Produce the line of planceer W Y, as shown by "W 1 Y'. Draw a duplicate of the plan, Q D A 1 in Fig. 
467, in such a manner that the diagonal line D X shall lie parallel to the horizontal line drawn through A, all 
as shown by Q' D 1 A 2 . At right angles to the line D' A", at any convenient point, as A 2 , draw the line A 2 C, 
in length equal to the distance A C in elevation, and through C draw a line parallel to D' A 2 , as shown by 
I X', cutting the diagonal line D 1 X 1 in the point X'. Then D 1 X 1 represents the diagonal plan of the Lip. 
From X 1 erect a perpendicular, X 1 M. which produce until it meets the line carried horizontally from the plan- 
ceer in the point B 1 . In like manner from D' erect a perpendicular, which produce until it meets the horizon- 
tal line L C in the point L. Connect L and B 1 , as shown. Then points in L B' correspond to points in A B of 
the elevation. Therefore at any convenient point, and at right angles to it, draw the line G H, upon which to 
construct a profile of the hip molding. Assume any point in the diagonal plan, as E, in the side D 1 A 2 , from 
which erect a line jierpendicular to D 1 X 1 , as shown by E F, which produce until it meets the horizontal line 
L C in the point L 1 , and thence carry it upward parallel to L B', cutting G H in the point F'. On either 
side lay off a space equal to F E of the diagonal plan, as shown by F 1 E l and F' E 2 . Through these points E 1 
and E 2 draw lines to K, being the intersection of the lines L B 1 and G II and a point corresponding to K" of 
the elevation. Upon these lines K E 1 and K E 2 , at proper distances from K, set off the edges of the hip mold- 
ing, as shown by E 3 and E 4 of the elevation. From K as center, with radius corresponding to the radius of the 




190 



Pattern Problems. 



profile in elevation, describe the shape of the roll, thus completing the profile of the hip molding in the diag- 
onal elevation. Space one-half of this profile, as K E 2 , in the usual manner, through the points in the roll of 
which carry lines parallel to L B l , cutting the line of planceer W T 1 , and through the points in the edges of 
which carry lines, also parallel to L B', until they meet the line of apron of the deck cornice, all as shown in 
the elevation. At any convenient point at right angles to the line L B 1 draw the straight line S E, upon which 




Fig. 468.— Elevation, Section, Diagonal Section and Development of the Patterns. 
Patterns for a Hip Molding on a Square Mansard Roof, Mitering Against a Bed Molding at the Top. 

lay off a stretchout of the profile in the usual manner, and through the jxfints draw measuring lines. "With 
the T-square parallel to this stretchout line, or, what is the same, at right angles to the lines of the molding in 
the diagonal elevation, and, bringing it successively against the points in "W 1 Y', and then against the apron of 
the deck cornice, as above explained, cut corresponding measuring lines drawn through the stretchout. Then a 
line traced through these points, as shown in the engraving, will be the pattern of the hip molding mitering 
against the horizontal planceer. 



Pattern Prohl 



ems. 19! 

576. Patterns for a Hip Molding on a Square Mansard Poof, Mitering Against a Bed Molding at the 
Top. — Let A C B in Fig. 4GS be the section of a mansard roof, the elevation of which is shown to the left of 
the section, and P E be any bed molding, the profile of which does not correspond to the molding nscd upon 
the hips. For the pattern of the hip molding to miter against this bed molding we proceed as follows : Since 
the angle of the roof is a right angle, the elevation may be used by which to construct a true face of the hip. 
JSTo other section than the original section will be recpiired for that purpose. It is necessary, however, to con- 
struct a diagonal section through the hip, in order to get the correct profile of the stay by which to place it in 
the elevation of the true face. At any convenient place lay off a plan of the roof, as shown by D 1 F D 2 in 
Fig. 469, and through this angle draw a plan of the hip, as shown by F Iv. From D 1 erect a perpendicular, 
D' C% in length equal to D C of the section. Through C 2 , parallel to D 1 F, draw C 2 K, producing it until it 
cuts the line representing the plan of the hip. From the points F and K in the lines representing the plan of 
the hip erect perpendiculars, as shown by F L and K C 5 . Draw L C 3 parallel to F K, as shown. From C 3 
erect a perpendicular C 3 E', in length equal to C E of the original section. 
Connect E 1 L. Then L C 3 E 1 will be a diagonal section of a portion of a roof, 
and L E 1 will be the length of the hip through that portion. At right angles 
to L E' draw M IF, upon which to construct a section of the hip molding. 
Take the point G in the line F D 1 at convenience, and from it erect a perpen- 
dicular to F K, cutting F K in the point II, and produce it also until it cuts 
the base line of the diagonal section L C 3 , as shown, and from this carry it 
parallel to the line L E', representing the jfitch of the hip, until it crosses the 
line M IF, cutting it in the point IF. Since D 1 F D 2 represents the angle over 
which the hip molding is to fit, and since G H is the measurement across that 
angle, if we set off from II 1 in the diagonal section a distance equal to H G, 
we shall have obtained a point by which the angle contained between the 
fascias of the hip molding may be determined. From IT on either side set off 
the distance II G of the plan, as shown by G 1 G\ Through these points draw 
lines representing the fascias of the hip molding, as shown by G 1 and G 2 . 
Add the fillets and draw the roll, all as shown. In the true face, Fig. -±68, draw 
a half section of the hip molding, as shown. M 2 H 3 corresponds to M II 1 of the 
diagonal section. Space this profile into any convenient number of parts 
in the usual manner, and through the points draw lines parallel to the lines of 
the hip molding indefinitely. Place a corresponding portion of the stay of the 
hip molding in the vertical section in which M 1 H 2 also corresponds to II 1 M in 
the diagonal section. Divide this section into the same number of equal parts, 
and through the points draw lines upward until they intersect with the profile 
of the bed molding, as shown from P 2 to B 2 . From the points in P 2 B 2 carry 
lines horizontally, intersecting the lines drawn from the profile in the true face. 
Then a line traced through these points of intersection will be the miter line 
between the hip molding and the bed molding, as seen in elevation. At right angles to the line of the hip 
molding, as shown in the true face, lay off a stretchout of the hip molding, as shown by S E, through 
the points in which draw the usual measuring lines. Place the T-square at right angles to the lines of the 
hip molding, and, bringing it successively against the several points in the miter line, as shown in eleva- 
tion, cut corresponding measuring lines, which will give that portion of the pattern shown from U to V. In 
like manner place the T-square against the point X in the true face, which is the point of junction between the 
flange of the hip molding and the apron of the bed molding corresponding to points 9 and 10 of the profile, 
and cut the corresponding measuring lines. The pattern is then completed by drawing a line from W to V 
and T to IT. 

577. The Patterns of a Hip Molding upon an Octagon Angle of a Mansard Poof Mitering Against an 
Inclined Wash at the Bottom.— -In Fig. 470, let D B represent the wash surmounting the base molding at the 
foot of a mansard roof, the inclination of which is shown by B A. Let E S T be the half profile of the hip 
molding which is required to miter against the wash D B, and let the angle of the roof upon which the hip 
molding occurs be an octagon angle, as shown by E G II in the plan. Problems of this nature are likely 
to reach the pattern cutter in various stages of completion, so far as relates to the drawings. They may 




\ 

Fig. 469.— Plan of Hip. 
Patterns for a Hip Molding on a 
Square Mansard Roof, Mitering 
Against a Bed Moldiwj at the 
Top. 



192 



Pattern Problems. 



present the plan correctly drawn, together with the elevation corresponding thereto, and a section, or 
nothing bnt the pitch of the roof, the angle of the miter and the profile of the hip molding may be 
given. Accordingly, in onr description we will start with the smallest number of given parts, and from 
them develop the several representations of the work, in order to afford the pattern cutter such knowl- 
edge as will enable him to start 
wherever circumstances may re- 
quire. Assume any point, A, 
in the pitch of a roof as a start- 
ing point by which to measure 
the angle of inclination. From 
A drop a vertical line, as shown 
by A 0, and from B, the point 
of intersection between the roof 
and the wash, draw a horizontal 
line cutting the vertical line in 
the point C. Draw a plan of 
the wash to an octagon angle, as 
shown by E G H I K F. Draw 
the miter line G Iv in plan. 
Show a top view of the hip 
molding as it would appear meet- 
ing this wash, by means of lines 
drawn parallel to the miter line 
G K, as shown by M L and N O. 
From the inside line of the wash, 
at any convenient point, as B 1 , 
set off B 1 C 1 , in length equal to 
B C of the section. Then the 
point C in the plan corresponds 
to both points A and C in the 
section. From the point C carry 
a line horizontally, or parallel to 
the line of plan, meeting the hip 
molding in any point, as P. 
From P and O draw vertical lines 
indefinitely, which intersect by 
horizontal lines drawn from the 
points A and B in the section. 
Connect the points of intersection 
between corresponding lines, as 
shown by the line P 1 0". Then 
P 1 O 1 will represent the inclina- 
tion of the hip molding as seen 
in elevation. The elevation may 
be completed by drawing the 
other lines, as shown. The ele- 
vation of hip thus obtained may 
be used in the following steps, or 
the plan itself from which the elevation was constructed may be used for that purpose. So far as cutting the 
pattern is concerned, it is not necessary to construct an elevation, or if the elevation be correctly given in the 
original drawings the patterns may be cut by it independent of the plan. Construct a section of the roof 
and wash, as though the roof were placed in a vertical position. Make A 2 B 2 equal to A B of the original 
section, and let the angle A 2 B 2 D 1 equal A B D of the original section. From the points A 2 and B 2 draw hor- 




Fig. 470.- 



-The Patterns of a Hip Molding upon an Octagon Angle of a Mansard Roof, 
Mitering Against an Inclined Wash at the Bottom. 



Pattern Problems. 



193 



izontal lines, which intersect by points dropped from P' and O 1 in the elevation, or from P and in the plan 
according to whichever is being used for the purpose. Through these points of intersection draw a line as 
shown by P 2 O 2 , which will represent the pitch of the hip, as seen in the plan of the roof, and which is to 'be 
used for measurement in the patterns. Complete the view of the hip by inserting one-half of the profile of 
the molding, as shown by R S T. 
Complete a corresponding view 
of the wash at the bottom by 
drawing lines from the point 
IP D 1 , all as shown. Divide the 
profile EST into spaces in the 
usual manner, and from the 
points carry lines parallel to the 
lines of the hip molding on to 
the wash indefinitely. Draw a 
duplicate profile in connection 
with the corresponding section, 
as shown by R 1 S 1 T 1 , which 
divide into the same number of 
parts, and from the points in it 
drop lines against the line of the 
wash, as shown by D 1 B 1 , and 
from the points in D' B 2 carry 
lines horizontally intersecting the 
lines dropped from the profile 
R S T. Then a line traced 
through these points of inter- 
section, as shown by R 2 S 2 T 2 , 
will be the miter line formed by 
the junction of the hip molding 
with the wash. 




Fig. 471. — The Patterns of a Hip Molding upon an Octagon Angle in a Mansard Roof, Mitering Against a Bed Molding of 

Corresponding Profile, 

to the line of the hip molding in the true face lay off a stretchout of the hip molding, as shown by U V. 
Through the points in it draw measuring lines in the usual manner. Place the T-square parallel to tin's stretch- 
out, or, what is the same, at right angles to the line of the hip molding, as shown in true face, and, bringing it 
successively against the points in the miter line R 2 S 2 T 2 , cut the corresponding measuring lines. Then a line 



194: 



Pattern Problems. 



traced through these points of intersection, as shown from W to Z, will be the cut to fit the bottom of the hip 



molding. 



578. The Patterns of a Hip Mold- 
iipon an Octagon Angle in a Mansard 
Poof, Miter ing Against a Bed Molding 
of Corresponding Profile. — This prob- 
lem, like that in Section 577, may reach 
the pattern cutter in drawings either 
more or less accurate, and in different 
stages of completion. Accordingly we give 
in this demonstration, so far as concerns 
drawing the elevation, a little more than 
is actually required for the development 
of the patterns. The drawings, as pre- 
pared by the architect or draftsman, may 
contain everything nec- 
essary to be used and 
ready for the develop- 
ment of the pattern, or 
they may contain the ele- 
ments from which the 
pattern cutter must con- 
struct such views as are 
necessary for him to use 
in the latter operation. 
In Fig. 471, let A B C 
represent the angle of 
the pitch of the roof, 
and let A D B be a sec- 
tion of the bed molding 
and apron finishing the 
mansard roof at the top. 
Let A B 1 be a continua- 
tion of the line of the 
planceer. Let & L B' 
be an octagon angle rep- 
resenting the plan of the 
hip over which the mold- 
ing fits. Let EFFE' 
be a profile of the hip 
molding, of which the 
portions E F and F 1 E 1 
correspond to the bed 
molding and apron, as 
shown from A to D. 
The pattern to be devel- 
oped is that of the hip 
molding mitering against 
the bed molding and 
apron A D. Commence 
by constructing a section 
of the roof, as shown by 
A A 2 C B 1 , in which 
draw a section of the bed molding and apron. From the several points iu the profile of the bed molding and 




Fig. 472.— Plan, True Face and Diagonal Section. 
The Patterns for the Miter at the Bottom of a Hip Molding on a Mansard Roof which is Octagon 

at the Top and Square at the Bottom. 



Pattern Problems. 



195 



apron carry lines vertically cutting the horizontal line A B'. Duplicate this line in plan in the same relative 
position, as shown by A 2 B 2 , making the several points in it correspond to the several points in A B 1 . Carry 
these lines horizontally indefinitely. Draw the miter line K L. Prolong the miter line K L, as shown by L M, 
upon which, in position corresponding to the position the molding is to occupy when upon the building, draw a 
profile of the hip molding, in which set off the points corresponding to the points in the profile of the bed 
molding and apron. From these points carry lines parallel to the miter line intersecting the lines drawn from 
the points in A" B\ By this means we have obtained a correct plan of the intersection between the hip mold- 
ing and the bed molding and apron. From the points in the plan obtained as just described drop lines verti- 
cally, and in turn intersect them by lines drawn from the several points in the section, as indicated by the 
dotted lines, in the engraving. By this means we produce an elevation of the junction between the hip mold- 
ing and bed molding corresponding to the plan already constructed. Next construct a section, placing it in 
a vertical position, instead of in an inclined position. Set off B 3 A 3 equal to B A 1 . Draw a section of the bed 
molding and apron, as shown by K 1 D 1 , 
corresponding to K D of the original 
section. From the points in the section 
K 1 D 1 carry lines horizontally, and inter- 
sect them by lines drawn from corre- 
sponding points in either plan or eleva- 
tion, according to which one is used, as 
indicated in the diagram. By this means 
is produced a representation of the true 
face of one half of the hip molding. 
In this true face insert a half profile, as 
shown by F 2 E 2 , which divide into any 
convenient number of spaces in the usual 
manner. Draw the miter line O P, rep- 
resenting the line of junction between 
the hip molding and the bed molding 
and apron. Inasmuch as the profile of 
the hip molding and that of the apron 
and bed molding correspond, this line 
O P is a straight line. Were they dif- 
ferent in profile it would be other than a 
straight line. From the points in the 
profile F 2 E 2 carry lines parallel to the 
line of the hip molding, cutting the miter line P, as shown. Inasmuch as point No. 1 in this profile falls out- 
side of the miter line P, a separate operation must be performed in order to obtain a measurement for its 
intersection. Draw as much of the profile of the hip molding as may be necessary in the vertical section, and 
from point No. 1 carry a line parallel to the lines of the section, intersecting the planceer line A 6 B', which is 
here inserted, being transferred from the other section for this purpose. From these points of intersection 
carry a line intersecting a corresponding line drawn from the same point in the profile F 2 E 2 , as shown by the 
point K 2 . Having thus obtained all the points in the miter line, for the pattern itself we proceed as follows : 
Lay off a stretchout of the complete molding, as shown by K S, placing the same at right angles to the lines of 
the hip, as shown in the true face, and through the points in the same draw miter lines in the usual manner. 
Place the T-square parallel to this stretchout line, or, what is the same, at right angles to the lines of the hip 
molding, and, bringing it successively against the points in the miter line, and also against the point K 2 , cut the 
corresponding measuring lines, as shown. Then a line traced through these points of intersection, as shown by 
Y T U W, will be the pattern sought. 

579. The Patterns for the Miter at the Bottom of a Rip Molding on a Mansard Poof tohich is Octagon at 
the Tojp and Square at the Bottom.— Bet L D B 1 in Fig. 472 be the plan of the roof at the base, and E-C G the 
plan at the top of the portion here made use of for the purpose of demonstration. Let A B C in Fig. 473 of 
the section indicate the pitch of the roof. Then, since it is square at the base and octagonal at the top, we have 




Fig. 473.— Section and Elevation. 

The Patterns for the Miter at the Bottom of a Hip Molding on a Mansard Roof 
tvhich is Octagon at the Top and Square at the Bottom. 



two converging 



hips, represented by Pt D and C 1 D, which unite and become a single profile at D. Let B P 



196 



Pattern Problems. 




of the section represent a wash, the plan of which is shown bj M IT P 1 of Fig. 472. Then the pattern required 
will be the shape of the hip molding to miter against this wash. But, since the two hip moldings join before the 
wash is reached, the pattern will be modified to the extent of fitting the inner edge of one against the corre- 
sponding edge of the other. This condition of things is shown in the elevation whicli is here introduced, not 
for any use it may be in the operation of cutting the patterns, but for more clearly showing the principle. The 
elevation is drawn by means of intersecting points from the section and the plan. "We are compelled to place 
the cut representing the elevation away from the plan in this instance, and, therefore, the connection between 
the two is not so clearly represented as it would otherwise be. D h IT B 3 corresponds to L> II B 1 in the plan. 
Horizontal lines from the points A B in the section are drawn, intersecting lines corresponding to the points 
already named. Let M L of the section represent one half of the profile of the molding whicli is required to 
be fitted to the converging hips. Our first step in the development of the patterns is in the construction of a 
section corresponding to the line of one of these hips as it appears in plan. Lay off D 1 C 2 equal to D C 1 of 
the plan, and from C 2 erect a perpendicular, C 2 A 1 , in length equal to A C of the original section. Connect 
A 1 I)'. Then A J C 2 D 1 is a section of the roof as it would appear if cut through on the line D C of the plan, 
and A 1 D 1 is the pitch of the hip. In order to locate the profile of the hip molding upon this section in correct 
position, take any point in the line R C, as G. Also lay off a corresponding point on the other arm, as G 1 . 
From G carry a line parallel to C 1 A 1 , producing it until it cuts the horizontal line drawn through A 1 at the top 
of the section, as shown by the point K 1 . From K 1 draw a line parallel to the pitch line A 1 D 1 . At any con- 
venient place in A 1 D 1 establish the point O. From the point O draw a line parallel to A 1 D' of convenient 

length. From the intersection of the 
line just drawn through O with the line 
from E?, set off the distance K G in the 
plan. In like manner from the point 
G 1 draw a line parallel to A 1 C 2 , cutting 
the line and the top in F 1 . From F 1 
draw a line parallel to A 1 D 1 , intersect- 
ing the line at in the point F 2 . From 
F 2 , on a continuation of the line "F 1 F s , 
set off a distance equal to F G 1 in the 
plan, as shown at G. Connect the point with G 9 and G 3 . Then the point of the jirofile will represent the 
corner of the sheeting boards over the hip. Construct a vertical section of the roof, placing the wash at proper 
angle with the same, In other words, make A 3 B 2 P 2 equal to A B P of the original section. By means of 
intersecting points from the vertical section just described and the plan, construct a true face of one of the hip 
moldings, as shown by Y S. Place a portion of the stay in this true face, locating it so that the point O 1 , which 
corresponds to of the hip section, shall fall upon the angle of the roof. Divide it into any convenient num- 
ber of spaces, numbering them in the usual manner. Fjom these points drop lines indefinitely through the 
face of the wash of the vertical section. Place also a part of the profile of the hip molding (greater than one- 
half) in proper position. From the points in this profile drop lines cutting the wash P 2 B 2 . From the points 
thus obtained carry lines horizontally crossing the true face, intersecting them with lines of corresponding 
numbers previously drawn. A line traced through the intersection of these points will give the pattern of the 
miter in the true face, all as shown by S T U. Note the points where this miter line intersects the miter line 
of the wash P 3 D 3 , which intersection carry back upon the profile L 2 M 2 , which in this case will correspond to 
the point 8. Locate the point 8 on the first section of the hip obtained at O, and use the remainder of profile 
S 14 for the other operation. Lay off a stretchout of the entire profile of the hip molding, as shown by "W.Y, 
through the points in which draw the usual measuring lines. "With the T-square placed at right angles to the 
lines of the hip, as shown in the true face, and brought against the points in the miter line STU, cut so many 
of the measuring lines drawn through the stretchout "W Y as correspond to those points. By this means that 
portion of the pattern shown by S 1 T 1 IP will be obtained. For the portion of the pattern corresponding to 
the part of the hip which miters against the other hip, we have first to construct a true face of the octagon side 
of the roof. To do this we require a diagonal section of the roof corresponding to the line D E in the plan. 
Lay off D 2 E 1 equal to D E of the plan, and from E' erect a perpendicular, E 1 A 2 , equal to C A of the section 
in Fig. 473. Connect A 2 and D 2 . Then A 2 D 2 is the length of the diagonal face of the roof measured on the line 
D E of the plan. Upon any convenient straight line lay off D' A 1 in Fig. 474, in length equal to D 2 A 2 , and 



Fig. 474.— Pattern of Miter between Hip Moldings near Base. 

The Patterns for the Miter at the Bottom of a Hip Molding on a Mansard Roof 

which is Octagon at the Top and Square at the Bottom. 



Pattern Problems. 



197 



from A 4 set off, at right angles to it, A 4 C s , in length equal to E C of the plan. Then D 4 A 4 C s shows in the 
fiat one-half of the diagonal face of the roof, or what is represented ty D E C in the plan. At right angles 




m , ^.-Patterns for a Hip Molding Mitering Against the Planceer of a Deck Cornice on a Mansard Roof, which 

at the Eaves is Square, at the Top Octagon. 

to D' C draw the remaining portion of the stay not used in connection with the true face, placing it in such a 

manner that the point 0', corresponding to of the hip section, shall fall upon the lme D' C , winch represents 

heTnde of the sheeting hoard. Through the point 8 of the section V W, corresponding to 8 of the secfaon 



198 Pattern Problems. 

L 2 M 2 , draw a line parallel to D* C, as shown by S" Y 1 . Then S 2 T 1 corresponds to S T of the true face. 
Space the profile L 5 M 7 into the same parts as used in laying off the stretchout "W Y, and through the points 
draw lines parallel to D 4 C 3 , cutting the line D 4 A 4 . From the points of intersection in the line D 4 A 4 , at right 
angles to S 2 "W, draw lines cutting S 2 Y', giving the points marked 8, 9, 10, 11, 12, 13 and 14. For convenience 
in using one stretchout for the entire pattern, transfer these points to the line S Y of the true face, and thence, 
at right angles to S Y, draw lines cutting the corresponding measuring lines of the stretchout. Then a line 
traced through these points of intersection, as shown from S 1 to X, will he the remainder of the pattern. 

580. Patterns for a Sip Molding Mltering Against the Planceer of a Peck Cornice on a Mansard Poof 
which at the Paves is Square, at the Top Octagon. — In Fig. 475 is shown the method of obtaining the miter 
against the planceer of a deck cornice formed by the molding covering a hip, which occurs between the main 
roof and that part which forms the transition from a square at the base to an octagon shape at the top. The 
roof is of the character sometimes employed upon towers which are square in a portion of their bight and 
octagon in another portion, the transition from square to octagon occurring in the roof. The hip mold- 
ing with which we have to deal covers what may be called a transition hip, being a diagonal line starting 
from one of the corners of the square part and ending at one of the corners of the octagon above. In the 
plan, F D indicates a line across the face of the transition part of the roof at a point somewhere between the 
top and bottom. D A 1 indicates a corresponding line through one of the adjacent sides of the roof. C A B is 
the angle of the pitch of the roof taken at right angles to one of the sides. A 1 C of the plan corresponds to 
A C of the section. D E of the plan represents the line of one of the hip moldings, and W L of the plan is 
the line through the transition part of the roof corresponding to A' C 1 of the principal parts of the roof. By 
means of intersection of lines drawn from corresponding points in the plan and the section already described, 
an elevation may be constructed, as shown by H N R P, if the same is desired. It is introduced here not for 
any service which it performs in connection with cutting the patterns, but to better explain the relationship of 
the several parts with which we have to deal. The first step in the development of the pattern is to construct 
a section of the roof as it would appear if cut through on one of the hip lines. In other words, to construct a 
section of the roof corresponding to D E of the plan. To do this proceed as follows : At any convenient 
place outside of the plan draw D 2 C 2 , in length equal to D E, and parallel to it. Erect a perpendicular, C 2 B 1 , 
in length equal to C D of the elevation. Connect B 1 D 2 , as shown. Then B 1 D 2 will be the length of the hip 
through that portion of the roof represented by the section constructed, and as shown by D E in the plan. The 
next step is to construct in connection with this hip section of the roof a true stay of the hip molding. To do 
this proceed as follows : Take any point, G, in the plan at a convenient distance from the angle WDA. 
Set off at the same distance from the angle on the opposite side G'. From G carry a line at right angles to and 
cutting E> 2 C 2 in the point IP, and from this point carry it parallel with the line D 2 B 1 indefinitely. At right 
angles to D 2 B 1 draw a line, as shown by Z H 1 , intersecting with the line last drawn from II 2 in the point H 1 . 
From H", along the line H 1 H 2 , set off a distance equal to II G of the plan. And from O in the line Z H 1 , cor- 
responding to O 1 of the plan, set off a distance equal to O 1 G 1 of the plan, as shown by O G s . Having by these 
points determined the angle of the hip molding finish, a representation of it is indicated in the drawing by 
adding the flanges in the roll. Since the miter required is the junction between the hip molding, the profile 
of which has just been drawn, against a horizontal planceer, the remaining step in the development of the pat- 
tern consists simply in dividing the profile into any convenient number of parts, and carrying points against 
the line of the planceer, as shown at B 1 , and thence carrying them across to the stretchout, as indicated. 
It is evident, however, upon inspection of the elevation, that the apron or fascia strip in connection with the 
planceer which miters with the flange of the hip molding, will form a different joint upon the side corres23ond- 
ing to the transition piece of the roof, than upon the side corresponding to the normal pitch of the roof. To 
obtain the lines for this miter an additional section must be constructed, corresponding to a center line through 
the transition piece, as shown by "W" L in plan. Prolong C 2 D 2 , as indicated, in the direction of "W 1 , and lay off 
"W 1 L', equal to W L of the plan. From L 1 erect a perpendicular, as shown by L 1 B 2 , equal to C B of the orig- 
inal section. Connect "W 1 and B 2 , against the face of which draw a section of the apron or fascia strip belong, 
ing to the planceer, as shown, and from the points in it carry lines parallel to B 2 B 1 until they intersect lines 
drawn vertically from the flange of the hip molding lying against that side of the roof, all as indicated by U X. 
From these points carry lines, cutting corresponding lines in the stretchout. Having obtained these points we 
then proceed. At right angles to the lines of the molding in the diagonal section lay off the stretchout of the 
hip molding S T, and through the points draw the usual measuring lines, as shown. Place the T-square at right 



Pattern Problems. 



199 



angles to the lines of the molding, or, what is the same, parallel to the stretchout line, and, bringing it succes- 
sively against the points formed by the intersection of the lines drawn from the hip molding and the planceer 
line B\ cut the corresponding measuring lines, as shown. In like manner bring the T-square against the points 
IT and X, above described, and "W and V, 
points corresponding with the opposite side 
of the hip molding, and cut corresponding 
lines. Then a line traced through these 
several points of intersection, as shown by 
IT 1 X' T 1 V 1 , will be the pattern sought. 

581. Patterns for a Ilij) Holding 
Mitering Against the Bed Holding of a 
Deck Cornice on a Mansard Poof which 
is Square at the Base and Octagonal at the 
Toj). — The problem presented in Fig. 476 
is similar to that described, with the differ- 
ence that a bed molding is introduced in 
connection with the planceer against which 
the hip molding is to be mitered. MEM 1 
represents a plan of the roof at the top, 
while L D M" represents a horizontal line 
at some point between the top and the bot- 



' llr+*fHB 4 -TG"''~ = -~ = -" = " 




torn. The intersection of the lines M L 
and E D prolonged would indicate the cor- 
ner of the building at the bottom of the 
roof, the structure being square at the base 
and octagonal at the top. The first step in 
the development of the pattern is to obtain 
a correct representation of the roof as it 
would appear if cut on the line D E. It is 
not necessary to take the entire length of 



Fig. 476.— Plan, Elevation, True Face and Pattern. 

Patterns for a Hip Molding Mitering Against the Bed Molding of a Deck 
Cornice on a Mansard Hoof, which is Sqxmre at the Base and Octagonal 
at the Top. 



the rafter, and therefore we construct a section of the roof corresponding to only so much of it as is indicated in 
the plan. At any convenient point lay off E 3 D% Fig. 477, ecpml to D E of the plan. From the point E s erect a 



200 



Pattern Problems. 



perpendicular, E 3 B 3 , in length equal to C B of the section of the roof. Connect B 3 and D 3 , which -will be the 
pitch of the hip corresponding to the line D E of the plan. Since we have constructed the section D 3 E 3 B' 
away from and out of line with the plan, it is necessary to draw a portion of the plan in immediate connection 
with the section. Lay off the angle I 1 II A 3 equal to the angle F D A 1 of the plan, and let A 3 C 3 equal A 3 C 
of the plan. Draw 11 C 3 , which corresponds to the hip line in plan. From the point H in the plan thus 
constructed lay off on either arm the points I and I 1 , equally distant from it and conveniently located for 
use in constructing the profile of the hip molding. From I' carry a line parallel to D 3 B 3 indefinitely. From 
the point I erect a perpendicular to H C 3 , cutting it in the point K, which prolong until it meets the base D 3 E 3 
of the diagonal section, from which point carry it parallel to the inclined line D 3 B 3 indefinitely in the direction 
of K'. At right angles to the inclined line D 3 B 3 draw a straight line, O 1 K', cutting the line last described in 
the point K 1 . From K 1 , measuring back on this line, set off the point I 3 , making the distance from E? to I 3 the 
same as from K to I of the plan. From O 1 in the line I 1 I 3 set off the distance O 1 I 3 equal to O I 1 of the plan. 
From these points I 3 and I 3 draw lines meeting the line O 1 K 1 at the point of its intersection with the line D 3 B'. 

Complete the profile of the hip molding, as indicated, laying off the 
width of the fascias from O 1 on these lines, adding the roll and edges. 
The next step in the development of the pattern is to draw a true 
face of the hip molding, which is done by transferring the section 
A B to a vertical position, as indicated by A 3 B 3 , Fig. 476, in connec- 
tion with which the bed molding against which the hip molding is to 
miter is also drawn, as shown. From the several points in this ver- 
tical section draw horizontal lines, which intersect by vertical lines 
dropped from corresponding points in plan. Then the line D 2 E' is 
the true face of one-half the hip corresponding to D E of the plan. 
In connection with the vertical section just described, place a half pro- 
file of the hip molding, a true section of which we have obtained by 
the process already explained, and place a duplicate of this portion of 
the profile in connection with the true face. Space both of these profiles 
into the same number of parts, and from the several points in each carry 
lines upward parallel to the two sections in which they appear, the lines 
from the profile in the vertical section cutting the bed molding, and the 
lines from the profile in the true face being continued indefinitely. From 
the points thus obtained in the bed molding carry lines horizontally, inter- 
secting those drawn from the profile in connection with the true face, 
Fig. .^.-section and Profile. producing the miter line, as shown by E 3 . By inspection of the plan 

Patterns for a Sip Molding Mttering ^ elevation ; t wm be seen ttat the "miter of the bed molding around 

Against the Bed Molding of a Deck , . . ° . 

Comiee on a Mansard Roof, which is ^e octagon at E is irregular. That is, its miter line does not coincide 
Square at the Base and Octagonal at with the line of the hip D E. If we divide the profile of the bed mold- 
the Top. ing iii the vertical section and also the profile of the bed molding, as 

shown in the plan, into any convenient number of parts, dropping points in the profile of the plan on to the 
miter line, thence carrying them downward and intersecting them with horizontal lines from the corresponding 
points of the bed molding in section, as shown at E 1 , we will have the appearance of the bed molding miter in 
elevation. By a similar operation the appearance of this miter in the true face could be obtained, but it has 
here been performed in the elevation, instead of in the true face, in order to avoid confusion of lines. Having 
obtained this line in the true face, its intersection with the miter line previously obtained at E 3 must be 
noted. A line from this point of intersection must then be carried parallel to the line of the molding back to 
the profile of the hip, and there marked, as shown by the figure 7^. The position of the point 7£ should now 
be marked upon the section of the hip molding previously obtained at O 1 . So much of the profile as exists 
between 1 and 7£ in the true face is used in obtaining the stretchout of this part of the pattern. The remain- 
ing portion of the stay, namely, from 7£ to 14, is afterward used for the true face of the octagonal side for 
the remainder of the pattern. At right angles to the line of the molding in the true face lay off a stretchout 
equal to that portion of the profile thus used, as shown by D N, through the points in which draw measuring 
lines in the usual manner. Place the T-square at right angles to the lines of the molding in the true face, and, 
bringing it against the several points in the miter line between the hip and bed molding at E 3 , cut correspond- 




Pattern Problem* 



201 



ing measuring lines drawn through the stretchout. Then a line traced through these points, as shown by S T, 
will be the miier line for that portion of the pattern corresponding to the part of the profile thus used. For 




Fig. 478.- The Pattern for a Hip Finish in a Curved Mansard Roof, the Angle of the Hip being a Right Angle. 

the other half of the hip molding, being that portion which lies on the face of the transition piece, another 



202 Pattern Problems. 

operation must be gone through. Construct a section of the roof corresponding to the line F G in the plan. 
At any convenient point lay off F' C 1 in Fig. 477, equal in length to F G. From the point C 1 erect a perpen- 
dicular, C IT, in length equal to C B of the vertical section. Connect F' and B 3 . Then F 1 B 3 is the length of 
the transition side of the roof through that portion corresponding to F G of the plan. By means of this section 
in the plan lay off an elevation of one-half of the transition side of the roof, by which to obtain the proper 
measurement of that flange of the hip molding lying against it, At any convenient point set off G 1 F 3 , 
in length equal to B 3 F 1 . At right angles to it set off G 1 E 4 , in length equal to G E of the plan, and F 3 D l , 
in length equal to F D of the plan. Connect F) 4 and E 4 . Then G 1 E 4 D" F 3 is an elevation of that portion of 
the roof represented by G E F> F in the plan. In connection with this elevation of the transition face of the 
roof, construct a vertical section of the roof as it would appear if cut on the line F G. In connection with the 
vertical section just described, place so much of the stay as was not used for the pattern already delineated, 
and in the representation of the elevation of the transitional face of the roof place a corresponding portion 
of the profile, each of which divide into the same number of spaces. From the points thus obtained cany 
lines parallel to the lines of the respective representations of the part, those in the vertical section cutting the 
bed molding, and those in the elevation being produced indefinitely. From the points in the bed molding 
of the vertical section thus defined carry lines horizontally intersecting those drawn from the profile in the ele- 
vation, thus establishing the miter line, as indicated at E 4 . At right angles to the line D 4 E 4 set off a stretchout 
of the profile, as shown by E P 3 , through the points in which draw the usual measuring lines. "With the 
T-square placed parallel to this stretchout line, or, what is the same, at right angles to the line D 4 E 4 , and, being 
brought successively against the points in the miter line, cut corresponding measuring lines, as shown. Points 
also are to be carried across, in the same manner as described, corresponding to the bottom of the apron or 
fascia strip in connection with the bed molding. Then a line traced through these points, as indicated by the 
line drawn from U to T', will be the pattern of the other half of the hip molding. By joining the two patterns 
thus obtained upon the center line of the stay corresponding to P T of the first piece or P 2 T 1 of the second 
piece, the pattern will be contained in one piece. 

5S2. The Pattern for a PCijy Finish in a Curved Mansard Poof, the Angle of the Hip leing a Bight 
Jungle, — The general features presented in the problem shown in Fig. 47S are similar to some of those already 
described. The parts requiring special attention are the flange strips, sometimes called sink strips, bounding the 
fascia of the hip molding, which in curved work must be cut in a separate piece, it being impracticable to turn 
them from the edges of the fascia. II K represents an elevation of a curved hip molding occurring in a roof, of 
winch E D is the vertical hight and W E? is a section. The first step to be described is the method of obtaining 
the pattern of the f ascias of the hip molding. For this purpose we have shown in the drawing such a represen- 
tation of it as would appear if the two fascias formed a close joint upon the angle of the roof, and we have sup- 
posed that the hip molding or the bead is to be added afterward on the outside over this joint. "We therefore 
consider the part to be dealt with the same as though it were the section of a molding, instead of a section of a 
roof, and the operations performed are identical with those employed in cutting a square miter. Space the pro- 
file into any convenient number of parts, introducing lines in the upper part in connection with the ornamental 
corner piece, shown by L F), at such intervals as will make it possible to take measurements required to describe 
the shape of it in the pattern. From this profile, by means of the points just indicated, lay off a stretchout, as 
shown by II 1 K 1 , and through the points draw the usual measuring lines. Bring the T-square against the several 
points in H K, and cut the corresponding lines drawn through the stretchout just described. Then a line traced 
through these points, as shown by IP K 3 , will be the outside line of the fascia. For the inside line take the 
given width of the fascia and set it off from this line, measuring at right angles to it, as indicated by A 1 B 1 , and 
not alone the measuring hues of the stretchout, as would be indicated by A 1 C. Then a line traced through 
these points, as shown from M 1 to L', will be the inside line of the fascia strip. The points in the ornamental cor- 
ner piece from L 1 to D 1 are to be obtained from the elevation, in case an elevation is furnished the pattern cutter, 
by measurement along the lines drawn horizontally through the several points in L D, and which are indicated 
in the stretchout line already referred to. Or the shape from L 1 to D 1 may be described arbitrarily at this stage 
of the operation, according to the finish required upon the roof. The latter plan is the correct one in principle. 
The method of constructing the elevation, working back from the profile thus established, is clearly indicated 
by the dotted lines in the engraving. Through the several points in the profile II K horizontal hues are drawn, 
as shown, and from the inside line of the j>attern of the fascia piece, as above described, lines are dropped, cut- 
ting these horizontal lines of corresponding numbers. Then a line traced through these points, shown from M 



Pattern Problems. 



203 



to L, will be the inside line of fascia piece in elevation. For the flange strip hounding the fascia piece, com- 
monly called the sink strip, an elevation of which is shown in the section from M 2 to D 2 , proceed as follows : 
Draw the line G F approximately par- 
allel to the upper part of the section 
M." D*, making it indefinite in length, 
which cut by lines drawn from the sev- 
eral points in M 2 W, at right 
angles to it, as shown. From 
F G, upon the several lines 
drawn at right angles to it, 
set off spaces equal to the 
distance upon lines of cor- 
responding number from 




I) E to the line M L of the elevation. Then a line traced through 
these points will represent the profile of this flange strip, as indi- 
cated by M 3 L 3 . In like manner set off in continuation of it the 
length measured upon the ornamental corner piece, all as shown 
by L 3 D 3 F. From this profile lay off a stretchout parallel to 
G F, as shown by M' D\ through the points in which draw meas- 
uring lines in the usual manner. Place the T-square parallel to 
this stretchout line, and, bringing it successively against points in 
both the inner and the outer lines of the elevation of the flange 
strip, as shown from W T>% cut the measuring lines of correspond- 



N 

Fig. 479.— Elevation, Plan and Diagonal Section. 
The Patterns for the Bead Capping a Hip Finish 
in a Curved Mansard Roof, the Angle of the Hip 
being a Bight Angle. 



204 



Pattern Problems. 



iuc number. Then lines traced through these points of intersection, as shown from M' to D 6 , will be the pat- 
tern of the flange strip bounding the edge of the fascia. 

583. The Patterns for the Bead Capping a Hip Finish in a Curved Mansard Roof, the Angle of the 
Hip being a Right Angle. — Let A E B in Fig. 479 represent the plan of a mansard roof or tower, the elevation 
of which is shown by H E K, over the hip of which a molding of any-given profile is to be fitted, in this case 
a three-quarter bead. Then the diagonal line E F in the plan represents the hip as it would appear if viewed 
from the top. At any convenient point parallel to E F, and ecpial to it, draw E 1 F', and from F 1 erect a perpen- 
dicular, F K 1 , in length equal to the vertical line in elevation E K. Divide E K and F 1 ~K l into the same num- 
ber of equal spaces. From the points in E K draw lines cutting the profile H K, as shown, and from the points 
thus obtained in H K drop lines vertically, producing them until they cut the diagonal line E F, as shown. 
Through the points in F 1 E? draw measuring lines in the usual manner, and intersect them by lines erected per- 
pendicularly to E F. Then a lino traced through these points of intersection, as shown by E 1 K 1 , will be the 
profile to which the molding covering the hip is to be raised. Inasmuch as the usual process of raising the 
curved molding requires for the adjustment of the machine, as well as for the description of the pattern, a knowl- 
edge of the center from which the curve is struck, divide the profile E 1 K 1 into such parts as will correspond to 

segments of circles. In this case the section from E 1 to L corre- 
sponds to an arc struck from the center M, and the section from 
L to K 1 corresponds to an arc strack from a center not shown in 
the engraving, but which will be found by the intersection of 
the lines L IS" and K? 1ST' produced. In Fig. 480 we show an 
enlarged section of the hip molding, including flanges and roll as 
it would appear at the bottom of the hip, and also another sec- 
tion as it would appear at the top. Upon inspection it is evident 
that the distortion to which these profiles is subjected is altogether 
owing to the change of direction in the hip molding. In other 
words, they are sections taken at right angles to the hip at differ- 
ent points, and therefore the angle in the one is a right angle cor- 
responding to the base of the roof, while the other is an obtuse 
angle corresponding to a section at right angles through the mold- 
ing at the top of the roof. The sections will be the same at all 
points if taken upon horizontal planes. The method of obtain- 
ing these several sections from the plan has been clearly described 
in connection with problems relating to hip finish upon straight 
mansard roofs, and therefore needs no further description at this time. 

5S4. The Patterns of the Hip Molding Finishing a Curved Mansard Roof which is Square at the Eaves 
and Octagonal at the Top. — The problem illustrated in Fig. 481 may be described as a combination of some of 
features of the last three problems presented. It is ordinarily presented, however, to the pattern cutter in a 
manner which requires the use of still other principles' than those we have explained, in order to develop 
the several shapes. CDEF represents the plan of the building at the base of the roof, while T6HW 
represents the plan of the roof at the top. It will be seen that the roof is square at the foot of the rafters 
and octagonal at the top. The same conditions may arise where the corners of the roof are chamfered, the 
chamfer being of unequal width, starting at nothing at the bottom and increasing to a considerable space at 
the top. D6I1E in the plan represents a chamfer of this kind, or a transition piece in the construction of a 
roof which, as above described, is square at the base and octagonal at the top. The same features are repre- 
sented in elevation by D 2 G 2 H 2 E\ The elevation is introduced here not for any use in pattern cutting, but 
simply to show the relation of parts. A B represents a section of the roof, showing the inclination and 
curve of the rafter. Space the profile B into any convenient number of parts, and from the points thus 
obtained draw horizontal lines indefinitely. Draw a duplicate section placed in a horizontal position, as shown 
by O 1 A 2 B 2 , which divide into like spaces, and draw lines from that horizontally cutting the hip line E H in 
plan, which becomes a miter line so far as the patterns are concerned. The intersections of lines drawn ver 
tically from the miter line E II with those drawn horizontally from* the profile O B, give the hue of the hip in 
elevation, as indicated by E 2 H 2 . Take a stretchout of the profile O B and lay it out at right angles to the 
horizontal lines drawn through the points in it, as shown by O 2 B 4 , through the points in which draw the usual 




Fig. 480.— Sections through Hip Finish. 
The Patterns for the Bead Capping a Hip Molding 
in a Curved Mansard Roof, the Angle of the Hip 
being a Right Angle. 



Pattern Problems. 



205 



measuring lines. Cut these measuring lines by lines drawn vertically from the points in E H. Then a line 
traced through these lines of intersection, as shown by E 3 II 3 , will be the line of the pattern corresponding to 




Fig. 4 8i.-m Pattern of the Hip Molding Finishing a Curved Mansard Roof wMch is Sauare at the Eaves and 

Octagonal at the Top. 

the line E II in the plan. For the width of the flange or fascia piece forming the , hip ^l^ 
described in Section 582. For the pattern of the transition piece we proceed as follows T jg^jjj 
of the transition piece, as shown in plan, draw the line P E. At any convenient place outside of the plan of 



206 



Pattern Problems. 



transition piece draw a duplicate of P E parallel to it, as shown by P 1 A 1 , and from the point A 1 erect a perpen- 
dicular, A' B 1 , in length equal to A B of the original section. In A 1 B 1 set off points corresponding to the points 
in A B, and through them draw horizontal lines, as shown. Place the T-square parallel to A 1 B 1 , and, bringing 
it against the points in E A, cut corresponding measuring lines. Then a line traced through these points of 
intersection, as shown by B' P 1 , will complete the diagonal section corresponding to P P in the plan. Of 
this diagonal section tahe a stretchout, B 1 P 1 , which lay off on the straight line corresponding to P H produced, 

all as shown by P 2 B 3 . Through the points in P 2 B 3 draw the usual 
measuring lines. "With the T-square placed parallel to this stretch- 
out bne, and brought successively against the points in E H, cut the 
measuring lines,- as shown. Then a line traced through these points 
of intersection, as shown by E' to H', will be one side of the required 
pattern. In like manner, having transferred points from E II 
to the corresponding line D G, cut the measuring lines from it, 
which will give the other side of the required pattern. By the 
same general means the shape of the panel occurring in the transi- 
tion piece is described in the pattern. In its original design it may 
have been drawn either in the plan or elevation, or it may have 
been designed in connection with both. Carry lines across it cor- 
responding to the points in E H, and by this means obtain measure- 
ment of its width upon lines corresponding to the lines drawn 
through the stretchout. Use the T-square for cutting these measur- 
ing lines in the usual manner. Then lines traced through the points 
of intersection, as shown by K' 1ST 1 M' L 1 , will be the pattern for 
the panel piece. The flange strips or sink strips are to be obtained 




same general method as described 
observed in this connection that in 



in Section 582. It 
the construction of 



Ml 



Fig. 482. — The Patterns for a Pedestal of which the Plan is an Equilateral Triangle. 

the elevation by means of the intersection of lines drawn from corresponding points in the section and plan, 
it is occasionally necessary to introduce other points than those first inserted, in order to obtain correspond- 
ing points of measurement in other representations of the parts. For instance, 6£ of the section A B corre- 
sponds to the lower point of the panel piece, as shown in elevation. A point is necessary to be inserted to 
locate this part. The same may be said of 10J and 11^, also shown in the same section. The reader will 
readily understand that in all profiles spaced in the manner employed in the roof here described, and, in 
fact, in almost all cases, additional points may be inserted at any time when found necessary. In cases where 



Pattern Problems. 



207 



greater accuracy is required in certain parts of the work than in others, the same end may be accomplished by 

inserting additional points in this general manner. 

5S5. The Patterns for a Pedestal of which the Plan is an Equilateral Triangle.— Let ABDC in Pig. 

482 be the elevation of a pedestal or other article of which the plan is an equilateral triangle, as shown by 

F E G. Construct the elevation so as to show one side in profile, 
and place the plan to correspond with it. Draw the miter lines 
E O and G 0. Divide the profile B D into spaces of convenient 
size in the usual manner, and number them as shown in the diagram. 
From the points thus obtained drop lines, cutting E and G 0, as 
shown. Lay off the stretchout X P at right angles to the side E G, 
and through the points in it draw measuring lines. Place the 
T-square at right angles to E G, and, bringing it successively against 
the points in the miter lines E O and G 0, cut the corresponding 
measuring lines. A line traced through these points will be the 
pattern, as shown by F± L M K. 

5S6. The Pattern for a Pedestal, Square in Plan. — In Fig. 
483, let A B D C be the elevation of a pedestal, the four sides of 
which are alike, being in 
plan as shown by E II G 
F, Fig. 484. The miters 
involved are what are 
called square miters, or 
miters forming a joint 
at 90 degrees. A square 
miter admits of certain 
abbreviations in the ope- 
ration of cutting it, 
which makes it peculiar 
as compared with oth- 
ers. In the case of mi- 
ters for all other angles 
the points must be first 
dropped from the eleva- 
tion on to the plan, cut- 

tino- the miter line, and then in turn transferred to the stretchout, 
which is laid off at right angles to the side of the plan. This is 
illustrated in the triangular pedestal just described, and also in the sev- 
eral polygonal shapes following this. A square miter may be cut 
in the same way, as is shown in Section 440, in which miters for 
several plans are obtained from the same profile. In practice, how- 
ever whether in the case of a four-sided article, as shown in the 
accompanying diagram, or in the case of a simple miter in a cornice 
or a gutter, the abbreviated method which is here illustrated is 
always used. This method, as will be seen, dispenses with the plan 
entirely. The plan E II G F, Fig. 484, is introduced only to show 
the shape of the article, and is not employed at all in cutting the 
pattern. Space the profiles, shown in the elevation by A C and BJD; 
in the usual manner, numbering the points 





Fig. 484.— Plan. 
The Pattern for a Pedestal, Square in Plan. 



Fig. 483.— Elevation. 
The Pattern for a Pedestal, Square in Plan. 



the points as shown. Set off a 
A right angles to the base line D of the pedestal, through the points in which draw meas 

uring lines, 

in the two profiles, cut the corresponding lines drawn throu, 
points, as shown by L I O S K, will be the pattern of a side, 
587. The Patterns for a Vase, the Plan of which is a 



Place the ^square parallel to the stretchout lines, and, bringing it successively against the points 

h the stretchout. A line traced through these 

Pentagon.— -In Fig. 485, let S C K T be the eleva- 



208 



Pattern Problems. 



tion of a vase, the plan of which is a pentagon, as shown by O C C ! E P. Construct the elevation in such a 
manner that one of the sides will be shown in profile. Draw the plan in line and in correspondence with it. 
Divide the profile into spaces of convenient size in the usual manner and number them. Draw the miter lines 
C II 1 and & IF in the plan, and, bringing the T-square successively against the points in the profile, drop lines 
across these miter lines, as shown by the dotted lines in the engraving. Lay off the stretchout M N at right 
angles to the piece in the plan which corresponds to the side shown in profile in the elevation. Through the 
points in it draw the usual measuring lines. Place the T-square parallel to the stretchout line, and, bringing it 
against the several points in the miter lines which were dropped from the elevation upon them, cut the corre- 
sponding measuring lines drawn through the stretchout. A line traced through the points thus obtained will 
describe the pattern. In the case of a complicated profile, or one of many different members, to drop all the 

points across one section of the plan C 1 II 1 II 2 C 3 woidd result in 
confusion. Therefore it is customary, in practice, to treat the pat- 
tern in sections, describing each of the several pieces of which it is 
composed independently of the others. In the illustration given 
we have divided the pattern at the point H, describing the upper por- 
tion from the profile and plan, as above, while the lower part is 
redrawn in connection with a section of the plan, as shown in Fig. 
486. Corresponding letters in each of the views represent the same 
parts, so that the reader will have no trouble in perceiving just what 
lias been done. Instead of redrawing a portion of the elevation and 
plan, as we have done in this case, various other methods are some- 
times resorted to by pattern cutters. It is considered best to work 
from one profile rather than to redraw a portion of it, as that always 
results in more or less inaccuracy. Therefore, after using the plan 
and describing a part of the pattern, as shown in the operation 
explained above, a piece of clean paper is pinned on the board, 
covering this plan and pattern, upon which a duplicate plan is drawn, 
from which the second section of the pattern is obtained. This ope- 
ration is repeated for each of the several sections of which the pat- 
tern is composed. As this method necessitates redrawing the plan 





Fig. 485.— Pattern for the Upper Part. 

The Patterns for a Vase, the Plan of which is a Pentagon. 

each time, which also leads to inaccuracies, some mechanics prefer, after getting one section of the pattern, to 
erase the points on the miter lines dropped from the elevations, thus adapting them for use again, and employ 
a fresh piece of paper only for the pattern. This method has the sanction of usage upon the part of some of 
the best pattern cutters in the country, and is probably quite as accurate as any. 

5S8. The Pattern for a Pedestal, the Plan of which is a Hexagon. — In Fig. 487, let C D F E be the ele- 
vation of a pedestal which it is desired to construct of six equal sides. Draw the elevation so that one of the 
sides will be shown in profile. Place the plan below it and corresponding with it. Divide the profile shown by 
the elevation into any convenient number of spaces in the usual manner, and, to facilitate reference to them. 



Pattern Problems. 



209 



Place the f -square parallel to the stretchout line, 



number them as shown Bring the T-square against the points in the profile and drop lines across one section of 
he plan, as shown by II X M. At right angles to this section of the plan lay off the stretchout line X O 

through the points m which draw the usual measuring lines. 

and, bringing it successively against the points in the miter 

lines H X and M X, cut the corresponding measuring lines, 

as indicated by the dotted lines. Then a line traced through 

the points thus obtained will be the required pattern as 

shown by P S T E. 

5S9. The Pattern for a Vase, the Plan of which is a 

Heptagon.— In Fig. 488, let E L P G be the elevation of 

the vase. Construct it in such a manner that one of its 

sides will be shown in profile. In line with it draw the 

plan, placing it so that it shall correspond with the eleva- 
tion. Space the profile L P in the usual manner, and from 

the points in it 
drop lines crossing 
one section of the 
plan, cutting the 
miter lines PS and 
H Y, as shown. 
Lay off a stretch- 
out, A P, at right 
angles to the side 
of the plan corre- 
sponding to the 
side of the vase 

shown in profile in the elevation. Through the points in it draw the 
usual measuring lines. Place the T-square parallel to this stretchout 
line, and, bringing it successively against the points in the miter lines, 
cut the corresponding measuring lines, as shown. A line traced 
through these points, as shown by K O W XT, will be the pattern of 
one of the sides of the vase. 

590. The Pattern for an Octagonal Pedestal— Let IUIGVL 
Fig. 4S9 be the elevation of a pedestal octagon in plan, of which 





Fig. 486.— Pattern for the Base. 
The Pattern for a Vase, the Plan of which is a Pentagon. 



Fig. 487. — The Pattern for a Pedestal, the Plan of which is a Hexagon. 

the pattern of a section is required. Draw the elevation in such a manner that one side will appear in profile 
in the elevation. Place the plan so as to correspond in all respects with it. Divide the profile G "W in the 
nsual manner, and from the points in it drop points upon each of the miter lines F T and P U in the plan. 



210 



Pattern Problems. 



Lay off a stretchout, B E, at right angles to the side of the plan corresponding to the side of the article shown 
in profile in the elevation, and through the points in it draw the usual measuring lines. Place the T-square par- 
allel to the stretchout line, and, bringing it successively against the points dropped upon the miter lines from 
the elevation, cut the corresponding measuring lines. A line traced through the points thus obtained will 
describe the pattern of one of the sides of which the article is composed. In cases where the profile is compli- 
cated, consisting of many members, and where it is very long, confusion will arise if all the points are dropped 
across one section of the plan, as above described. It is also quite desirable in many cases to construct the pat- 
tern of several pieces. In such cases various methods are resorted to, several of which are fully described in 
connection with the problem showing a pentagon plan (Section 5S7). In the present case the pattern is con- 
structed of two pieces, being divided at the point S of the profile. The lower part of the pattern is cut from 

the plan drawn below the elevation, while the upper part of the pat- 
tern is cut by means of a part of the plan redrawn above the eleva- 
tion, thus allowing the use of the same profile for both. The same 
letters refer to similar parts, so that the reader will have no difficulty 
in tracing out the relationship between the different view's. 

591. The Patterns of a Fmial, the Plan of which is Octagon with 
Alternate Long and Short Sides.— In Fig. 490, letALMNOPRST 
be the elevation of the fmial corresponding to the plan which is shown 
immediately below it. The elevation is so drawn as to show the pro- 
file of one of the long sides, for the pattern of which proceed in the 
usual manner. Divide the profile A L M 1ST O P into any number 
of convenient spaces, as shown by the small figures, and from the 
points thus obtained drop lines across the corresponding section in the 
plan, cutting the miter lines DEC and D 2 F C, as shown. To 
save space, a duplicate section of the plan is shown below by 
by E' C a F, and in the demonstration C 5 E 1 and C F 1 are to be con- 




Fig 



-The Pattern for a Vase, the Plan of ivhich is a Heptagon. 



sidered the same in all respects as C E and C F. The same applies to the stretchout lines, which are indicated 
by the same letters. Perpendicular to D D 2 lay off a stretchout, as shown by G H, through the points in which 
draw measuring lines in the usual manner. Place the T-square parallel to the stretchout line, and, bringing it 
against each of the several points in D E C and D 2 F C, cut the corresponding measuring lines. Then a line 
traced through these points of intersection will be the pattern sought. For the pattern of the short sides a 
somewdiat different course is to be pursued. A profile of the piece as it would appear if cut on the line C D 
must first be obtained. To do this proceed as follows : From the points in C E dropped from the profile carry 
lines parallel to E K across C D, cutting C K, as shown. At any convenient place lay off B 1 P 1 , Fig. 491, in length 
equal to C D of the plan. On B 1 erect the perpendicular B 1 A 1 , equal to B A of the elevation. On B 1 P 1 lay off 
points corresponding to the points obtained in C D of the plan, as above explained, and for convenience in the 
succeeding operations number them to correspond with the numbers in the profile from which they are derived. 
From the several points in the profile of the elevation draw horizontal lines, cutting the central vertical line 
A B, as shown. Set off points in A 1 B 1 in Fig. 491 to correspond, and through these points draw horizontal 



Pattern Problems. 



211 



lines, which number, for convenience of identification, in the following steps. From the several points in B' P" 
cany lines vertically, intersecting corresponding horizontal lines. Then a line traced through these points, as 
shown by A 1 L 1 M 1 W O l P 1 , will be the profile of the short side on the line C D of the plan? After obtaining 
the profile as here described, for the pattern of the short side proceed as follows : Perpendicular to K E of the 




The Pattern for an Oetaaonal Pedestal. 



short side lay off a stretchout of the diagonal profile, as shown by C" E", through the points in which draw 
measuring lines in the usual manner. Place the T-square parallel to the stretchout line, and, bringing it against the 
several points in the miter lines DEC and DEC bounding the short side in the plan, cut the corresponding 



measuring lines. 



Then a line traced through these points, as shown in the diagram, will be the required pattern. 



212 



Pattern Problems. 



592. The Pattern for a Newel Post, the Plan of which is a Decagon.— In Fig. 492, let V WU'SP OET 

be the elevation of a newel post which is required to be constructed in ten parts. Draw the plan below the 




Tlie Patterns of a Finial, the Plan of which is Octagon with Alternate Long and Short Sides. 

elevation, as shown. The elevation must show one of the sections or sides in profile, and the plan must be 
placed to correspond with the elevation. Space the molded parts of the profile in the usual manner, and from 



Pattern Problems. 



213 



w 



the points in them drop lines crossing the corresponding section of the plan, as shown by G X H, and cutting the 
two miter lines G X and II X. Lay off the stretchout line C D at right angles to G H, and through it draw the 
customary measuring lines. Place the T-square parallel to the stretchout, and, bringing it against the several 
points in the miter lines G X and II X, cut the corresponding measuring 
lines. A line traced through the points thus obtained will describe the pat- 
tern. In order to avoid confusion of lines, which would result from drop- 
ping points from the entire profile across one section of the plan, a duplicate 
of the cap A 1 "W 1 is drawn in Fig. 493 in connection with a section of the 
plan, as shown by G 1 X 1 IP, which are employed in precisely the same man- 
A ner as above described, thus completing 

th. 1 pattern in two pieces, the joint being 
formed at the point numbered 11 of the 
profile and the stretchout. 

593. The Pattern, for an Urn, the 
Plan of which is a Dodecagon. — In Fig. 
494, let X A G H be the elevation of an 
urn to be constructed in twelve pieces. 
The elevation must be drawn so as to show 
one side in profile. Construct the plan, as 
shown, to correspond with it and draw the 
miter lines. Divide the profile A S G 
into spaces in the usual manner, and from 
the points thus obtained drop lines across 
one section, X X of the plan. Lay off 
the stretchout C D at right angles to the 
side X O of the plan. Place the T-square 
parallel to the stretchout, and, bringing it 
successively against the several points in 
the miter lines X X and O X, cut the cor- 
responding measuring lines. A line traced 
through the points thus obtained will de- 
scribe the pattern sought. In this illustra- 
tion we have shown a method sometimes 
resorted to by pattern cutters to avoid the 
confusion resulting from dropping all the 

points across one section of the plan. The points from 13 to 20 inclusive are 
dropped upon the line X. The stretchout C D is drawn in exactly the mid- 
dle of the pattern. Points are transferred by the "["-square from X to the 
measuring lines on one side of the stretchout, the points on the other side being 



nO 



I 



I2l] 



Elev. 




147 4- 16 

5 17 

-Diagonal Section. 



B' 

Fig. 491.- 

The Patterns of a Finial, the Plan of 
which is Octagon with Alternate 
Long and Short Sides. 



23 p 




Plan 

Fig. 49a.— Elevation, Plan and Pattern of Lower Part. 
The Pattern for a Newel Post, the Plan of which is a Decagon. 

obtained by duplicating distances from D on the several lines. The points 1 to 13 are dropped on NX only. 
The stretchout E F is laid off at right angles to M X and directly in the middle of the pattern, and the T-square 
being set parallel to E F, the points are transferred to the measuring lines on one side of E I, while the distances 



214 



Pattern Problems. 



Twelve Pieces.- 
A' 



on the opposite side are set off by measurement, as described above in the first instance 
found advantageous in complicated and very extended profiles. 
594. The Patterns for an Elliptical Yase Constructed 
ellipse, by whatever rule is most convenient, of the length and 
breadth which the vase is required to have. Draw the sides 
of the vase about the curve, as shown in Fig. 495, in such a 
manner that all the points X, T, Z, etc., shall have the same 
projection from the curve. Complete at least one-fourth of 
the plan by drawing miter lines, as shown by P C, M C, O C, 
U C and Iv C. Above the plan construct an elevation of the 
article, or over one end draw a profile simply, as shown by 
H V W L. Only the profile of the elevation is needed for 
the purpose of pattern cutting, but the other lines are desira- 
ble in process of de- 



This plan will be 
The first step is to draw an 



signing, in order that 



the effect may be con- 
sidered before the ar- 
ticle is constructed. 





site in obtaining a profile of the third section, 
its other miter line C P. From C draw CD at 



F'S- 493.— Pattern of Cap. 
The Pattern for a Newel Post, the Plan of which is 
a Decagon. 

Divide the profile H T ¥ L in the usual manner, and from the seve- 
ral points in it drop lines across the corresponding section (So. 1) of 
the plan. Take the stretchout of H V "W L and lay it off at right 
angles to the side of section JSTo. 1 of the plan, as shown by E F. 
Through the points in it draw the customary measuring lines. Place 
the T-square parallel to this stretchout line, and, bringing it against the 
several points dropped upon the miter lines N C and U C bounding 
No. 1 of the plan, cut the corresponding measuring lines. Then a line 
traced through the points thus obtained will be the pattern of section 

No. 1. Across the sec- 
ond section in the plan, 
from the points already 
obtained in TJ C, draw 
lines parallel to 1ST, the 
side of it, and produce 
them until they meet 
A C,- which is a line 
drawn from C at right 
angles to U produced. 
Then the points in A C 
serve to obtain a profile 
of the section numbered 
2. In like manner con- 
tinue the points from 
C across the third sec- 
tion in the plan, also par- 
allel to M, the side of 
it, and produce them un- 
til they cut C B, which 
is a line drawn from C 
at right angles to II 
produced. Then C B 
contains the j)oints requi 



V Plan yy 

Fig. 494. — The Pattern for an Urn, the Plan of which is a Dodecagon. 



Continue the points in C ~M across the fourth section, cutting 
angles to the side P M of the section. Then upon C D 



right 



Pattern Problems. 



215 



being cut by the lines drawn across the section, will be found the points necessary to determine the profile of the 
fourth pattern. Produce the line of the base of the elevation indefinitely, as shown by C 1 C C 3 , and also the line 
of the top A' E = D 3 . From the several points in the profile II Y "W L draw lines indefinitely, parallel to the 




Fir,, A9S .—Thc Patterns for an Elliptical Vase Constructed in Twelve Pieces. 



lines iust described and as shown in the diagram. From C, upon the base line produced, set off points corre- 
sponding to the points in C A of the plan, making the distance from C in each instance the same as the dis- 
tance from C in the plan. Number the points to correspond with the numbers given to the points m the pronle 



216 



Pattern Problems. 



II V "W L, from which they were derived. In like manner from C set off points corresponding to the points 
in C B of the plan, numbering them as above described. From C s set off points corresponding to those in 
CD of the plan, likewise identifying them by figures in order to facilitate the next operation. From C erect 

the perpendicular C A 1 ; likewise from C 2 and C a erect the perpendicu- 
lars C 2 B 2 and C 3 D 3 . From each of the points laid off from C, and also 
from each of those laid off from C 2 and C s , erect a perpendicular, pro- 
ducing it until it meets the horizontal line drawn from the profile II V 
"W L of corresponding number. Then lines traced through these several 
intersections will complete the profiles, as shown. Perpendicular to the 
side of each section in the plan, lay off a stretchout taken from the pro- 
file corresponding to it, just described, and through the points in the 
stretchout draw measuring lines in the usual manner, all as shown by 
E 1 F'j E 2 F 2 and E 3 F 3 . Place the T-square parallel to each- of these 
stretchout lines in turn, and, bringing it against the several points in the 
miter lines bounding the sections of the plan to which they correspond, 
cut the measuring lines in the usual manner. Then lines traced through 
the points of intersection thus obtained, all as shown in the diagram, will 
complete the patterns. 

595. The Patterns for a Prop upon the Face of a Bracket. — In 
Figs. 496 and 497 methods of obtaining the return strip fitting around 
a drop and mitering against the face of a bracket, are shown. Simi- 
lar letters in the two figures represent similar parts, and the following 
demonstration may be considered as applying to both. Let A B D C be 
the elevation of a part of the face of the bracket, and H Iv L a portion 
of the side, showing the connection between the side strip of the drop 
E F G and the face of the bracket. Divide the profile F G into any 
convenient number of parts in 
the usual manner, as shown by 
the small figures. Produce 
1ST K, as shown by P, and on 
O P lay off a stretchout, through 
the points in which draw the usual measuring lines. From the points 
in the profile F G carry lines at right angles to the bracket, intersect- 
ing the profile of the face N M, against which the drop is to miter. 
Reverse the T-square, placing the blade parallel to the stretchout line 
O P, and, bringing it successively against the points in N M, cut the 
corresponding measuring lines, as indicated by the dotted lines. Then 
a line traced through these several points of intersection, as shown by 
R P, will be the pattern of the strip fitting around E F G and 
mitering against the irregular surface 1ST M of the bracket face. 

596. The Patterns of a Boss Fitting over a Miter in a Molding. 
—Let ABCin Fig. 498 be the part elevation of a pediment, as in a 
cornice or window cap, over the miter in which, and against the mold- 
ing and fascia, a boss, F K G H, is required to be fitted, all as shown 
by A D E. For the patterns we proceed as follows : Divide so much 
of the profile of the boss K F II G as comes against the molding, 
shown from K to F, into any convenient number of parts, and from 
these points draw lines parallel to the lines of the molding until they 
intersect the profile of the molding, as shown from N to 0. Also 
draw a line from the point II until it intersects the fascia in the point E. Then the points from N to O and 
the point E are the points by which measurements are to be taken in laying out the pattern on the stretchout 
line. In line with the side elevation lay off a stretchout of the boss, as shown by K' K 2 , dividing the portion 
K 1 F 1 , which corresponds to K F of the elevation, into like spaces, through which draw the usual measuring lines. 




Fig. 



496. — The Patterns for a Drop upon, 
the Face of a Bracket. 




Fig. 



497. — The Patterns for a Drop upon the 
Face of a Bracket. 



Pattern Problems. 



217 



In the same manner divide the space G 1 K\ uhich corresponds to G K of the elevation, into the same number 
of spaces as employed m the portion F K, to obtain points in the profile of the molding X 0, and also draw 
measuring lines, as shown. Place the T-square parallel to the stretchout line K 1 K% and, bringing it against the 
several points in N O, cut corresponding measuring lines, as shown. Then lines traced through these & points of 
intersection, as shown by K 1 L and M K% will be the required pattern. 

597. The Putt, rns of an Octagonal Shaft, the Profile of which Is Curved, Mitermg upon the Ridge of a 
Roof. —In Figs. 499 and 500 are shown the elevation and patterns of a tinial of a character somewhat common 
in cornice work. The shaft is octagon in shape. Four crockets and a point constitute the flower surmounting 
the same. The neck molding immediately below the flower consists of eight simple octagon miters, the pat- 
terns for which are cut by the ordinary rule, and need not 
be described in this connection. The shaft below the neck 
molding miters over the ridge of the roof. It is also curved 
in its profile, and by reason of these several combined fea- 
tures presents conditions differing from other problems of a 
similar character already demonstrated. For the patterns 
proceed as follows : Construct a plan of the shaft at its 
largest section, as shown by A, B, C, etc., from the center 
of which to two of the angles draw miter lines, as shown 
by G II and G II. Divide the profile of the side of the 
shaft J L into any number of parts in the usual manner, 
and from these points carry lines vertically crossing the 
miter lines G IT and G II. Bisect the section bounded by 
the miter line, as shown by E 1 F 1 , upon which line lay off a 
stretchout of the profile J L, drawing measuring lines 
through the points. Place the T-square parallel to the 
stretchout line, and, bringing it successively against the 
points in G H and G II, cut corresponding measuring lines, 
as shown, and through the points thus obtained trace lines, 
all as indicated in the drawing. This gives the general 
shape of the pattern for the sections of the shaft. By 
inspection of the plan and elevation together, it will be 
seen that to fit the shaft over the roof some of the 
sections composing it will require different cuts at their 
lower extremities. Two of the sections will be cut the 
same as the pattern already described. They correspond to 
the side marked A in the plan and the one opposite. Two 
others, one of which is indicated in the plan by C, and 
which is also shown in the elevation by n m n, will be 
cut to fit over the ridge of the roof. The remaining four 
pieces will be cut to fit against the pitch of the roof, 
as shown by n o in the elevation, and corresponding to the sides, of which B in the plan is one. For the 
sections corresponding to the one shown in the center of the elevation proceed as follows : From so many 
of the points in the profile T T as occur below a point opposite the ridge of the roof m, draw lines at 
right angles to the center line of the shaft, crossing the lines K I, K I, representing the pitch of the roof, 
ail as shown. Thus it will be seen that the line drawn from 4 touches the ridge in the point m, while the line 
drawn from 3 corresponds to the point at which the side terminates against the pitch of the roof. Therefore, 
in the pattern draw a line from the center of it, on the measuring line 4, to the side of it on the measuring 
line 3, all as shown by m' n l and w> n\ Then these are the lines of cut in the pattern corresponding to m n 
and m n of the elevation. By inspection of the elevation, for the remaining four sides it will be seen that it 
is necessary to make a cut in the pattern from one side, in a point corresponding to 3 of the profile, to the other, 
in a point corresponding to 1 of the profile, all as shown by n o. Talcing corresponding points, therefore, in 
the measuring lines of the pattern, draw the lines n' o\ as shown. Then the original pattern, modified by cut- 
ting upon these lines, will constitute the pattern for the other sides. In this connection we may remark that 




Fig. 4QS-- 



The Patterns of a Boss Fitting over a Miter 
in a Molding. 



218 



Pattern Problems. 







1 r~m^ I yf r 



^asusa 



Fig. 500.— Elevation of Shaft enlarged, 
also Plan and Pattern. 



for the crockets and point no pattern can be described. Of course an approximation to the forms shown might 
be devised, consisting of geometrical shapes, but it is far better in point of construction, wherever possible, to 
use either pressed work or hand-hammered work instead. Therefore we make no attempt to show patterns 
for the foliated parts. 

59S. To Construct a Ball in any Number of Pieces, of the General Shape of Zones. — In Fig. 501, let 
A O G II be the elevation of a ball which it is required to construct in thirteen pieces. Divide the profile into 
the required sections, as shown by 0, 1, 2, 3, 4, etc., and through the points thus obtained draw parallel horizon- 
tal lines, as shown. The divisions in the profile are to be obtained by the following general rule, applicable in 
all such cases : Divide the whole circumference of the ball into a number of parts equal to two times one less 
than the number of pieces of which it is to be composed. In convenient proximity to the elevation, the centers 
being located in the same line, draw a plan of the ball, as shown by K M L 1ST. Draw the diameter K L paral- 
lel to the lines of division in the elevation. "With the T-square placed at right angles to this diameter, and 

brought successively against 
the points in the elevation, 
drop corresponding points 
upon it, as shown by 1, 2, 
3, 4, etc. Through each of 
these points, from the center 
by which the plan is drawn, 
describe circles. Each of 
these circles becomes the 
plan of one edge of the belt 
in the elevation to which it 
corresponds, and is to be 
used in establishing the 
length of the arc forming 
the pattern with which it 
corresponds. Through the 
elevation, at right angles to 
the lines of the zones or 
belts of. which the ball is 
to be composed, draw a 
diameter, as shown by G A, 
which produce in the direc- 
tion of indefinitely. Con- 
struct chords to the several 
arcs into which the profile 
is divided by the division 
lines, which produce until 

they cut G A O, as shown by 1 2 E, 2 3 D, 3 4 C, 4 5 B and 5 6 A. Then E 2 and E 1 are the radii of parallel 
arcs which will describe the pattern of the first division above the center zone, and D 3 and D 2 are the radii 
describing the pattern of the third zone, and so on. From E 1 in Fig. 502 as center, with E 2 and E 1 as radii, 
strike the arcs 2 2 and 1 1 indefinitely. Step off the length on the corresponding plan line, and make 1 1 equal 
to the whole of it, or a part, as may be desired — in this case a half. In like manner describe patterns for the 
other pieces, as shown, struck from the centers D', Fig. 503 ; C\ Fig. 504 ; B 1 , Fig. 505, and A 1 , Fig. 506. The 
pattern for the smallest section, as indicated by F in the plan, may be pricked directly from it, or it may be 
struck by a radius equal to F 6 in the plan. The center belt or zone, shown in the profile by 1 0, is a flat band, 
and is therefore bounded by straight parallel lines. The width is taken by 1 in the elevation, and the length 
is measured upon 1 of the plan, all as shown in Fig. 507. 

599. To Construct a Ball in any Number of Pieces, of the General Shape of Gores. — Draw a circle of a 
size corresponding to the required ball, as shown in Fig. 50S, which divide, by any of the usual methods em- 
ployed in the construction of polygons, into the number of parts of which it is desired to construct the ball, in 
this case twelve, all as shown by E, F, G, II, etc. From the center draw miter lines, as represented by B E 



Fig. 499.— Elevation of 
Finial, of which the 
Shaft is a Part. 




The Patterns of an Octagonal Shaft, the Profile of which is Curved, Mitering upon the Ridge 

of a Roof. 



Pattern Problems. 



219 



and E F. If the polygon is inscribed, as shown in the illustration, it will be observed that the arc of the circle, 
as, for example, U C, does not form a profile in dimensions corresponding to the middle line of the sections of 
which the ball is to be constructed. Hence, it is necessary to draw a new profile, which may be done with suf- 
ficient accuracy for all practical purposes by taking the radius of the profile, and a point for the center whoso 
distance from the line A V prolonged is equal to the distance from the point IP to U in the plan. Then, from the 






Fig. 504.— Pattern of 
Zone 3 4. 



Fig. 502.— Pattern of Zone 1 2. 





Fig. 505.— Pattern of 
Zone 4 5. 



>tf 




Fig. 503. — Pattern of Zone 2 3. 



Fig. 506.— Pattern of 
Zone 5 6. 



Fig. 501.— Plan and Elevation. Fig. 507.— Pattern of Middle Zone. 

To Construct a Ball in any Number of Pieces of the General Shape of Zones. 

point located near V, as above described, as center, and with a radius equal to R U, strike the arc B A, which 
forms the profile of a section of the ball on its center line. Divide B A into any convenient number of equal 
parts, and from the divisions thus obtained carry lines across one of the sections at right angles to a line drawn 
through its center, and cutting its miter lines, all as shown in K E and R F. Prolong the center line R C, as 
shown by S T, and on it lay off a stretchout obtained from B A, through the points in which draw measuring 
lines in the usual manner.' Place the T-square parallel to the stretchout line, and, bringing it successively 
against the points in the miter lines R E and R F, cut the corresponding measuring lines, as shown. A line 
traced through these points will give the pattern of a section. If, on laying out the plan of the ball, the poly- 



220 



Pattern Problems. 



M L 



gon had been drawn about the circle, instead of inscribed, as shown in the engraving, it is quite evident 
that a quarter of the circle would have answered the purpose of a profile. These points, with reference to the 

profile, are to be observed in determining the size of the ball. 
In the illustration presented, the ball produced will correspond 
in its miterdines to the diameter of the circle laid down, while 
if measured on lines drawn through the center of its sections it 
will be smaller than the circle. 

600. The Patterns of a Square Shaft to Fit Against a 
Sphere. — In Fig. 509, let H A A' K be the elevation of a 
square shaft, one end of which is required to fit against the 
ball D F E. From the center G describe the circle of the ball. 
Through G draw a vertical line, as shown by F L. At equal 
distance from either side of this center line F L, draw the sides 
of the shaft, as shown by H A and K A'i continuing them 
across the line of the circumference of the ball indefinitely. 
From the points of intersection between the sides of the shaft 
and the circumference of the ball, A or A 1 , draw a line at 
right angles to the sides of the shaft, across the ball, cutting 
the center line, as shown at B. Set the dividers to G B as 
radius, and from G as center, describe the arc C C Then 
HCC'K will be the pattern of one side of a square shaft to 
fit against the given ball. 

601. To Descrlhe the Pedtern of an Octagon Shaft to Fit 
Against a Ball. — Let H F Iv in Fig. 510 be the given ball, of 
which G is the center. Let ~D~ C 2 C 3 D 3 E represent a plan of 
the octagon shaft which is required to fit against the ball. 
Draw this plan in line with the center of the ball, as indicated 
by F E. From the angles of the plan draw lines indefinitely, 
cutting the circle. From the point A or A 1 , where the side 
in profile cuts the circle, draw a line across the center line of 
the ball F E, cutting it in the point B, as shown. Through 
B, with the center by which 




Fig. 508. — To Construct a Ball in any Number of 
Pieces, of the General Shape of Gores. 



the circle of the ball was 

struck, describe an arc, cut- 
ting the two lines drawn from the inner angles C C 3 of the plan, as shown by 
C and C. Then ICC'N will be the pattern of one side of an octagon shaft 
mitering against the given ball H F K If it be desired to complete the 
elevation of the shaft meeting the ball, it may be done by carrying lines 
from C and C 1 horizontally until they meet the outer line of the shaft in 
the points D and D 1 . Connect C and D 1 , also C and D, by a curved line, 
the lowest point in which shall touch the horizontal line drawn through B. 
Then the broken line D C C D 1 will be the miter line in elevation formed 
by an octagon shaft meeting the given ball. 

602. Patterns for the Volute of a Capital.— Dr&v? an inverted plan of 
the parts, as shown in Fig. 511, and through the center of one of the volutes 
draw a line, A B, which shall correspond to the center line of the patterns. 
Construct the diagonal elevation, as shown, placing it in correspondence with 
the plan. Divide the volute, as shown in the diagonal elevation, into any 
convenient number of parts, numbering them for convenience of identifica- 
tion, as shown. From each of the several points in the elevation thus 
obtained drop lines crossing the plan, as shown. Prolong the line A B, as shown by B C, upon which lay off 
the stretchout of the several parts of which the volute is composed, drawing the usual measuring lines. Place 
the T-square parallel to the stretchout line, and, bringing it against the points formed by the lines of the eleva- 




Fig. 509. — The Patterns of a Square 
Shaft to Fit Against a Sphere. 



Pattern Problems. 



221 



tion crossing the plan, cut the corresponding measuring lines drawn through the stretchout line, all as shown. 
In order to avoid confusion of lines, but one-half of each pattern is shown in the engraving. In ordinary 
work sufficient accuracy is obtained if the sides of the volute are pricked f 

directly from the diagonal elevation, which saves the long and tedious ope- 
ration recpiired to develop them. It will be«seen, by inspection of the eleva- 
tion and plan, that the difference in the length of the sides, as shown in the 
two views, is very slight indeed. 

603. — Hie Patterns for a Cornucopia in Eight Pieces. — In Fig. 512 
is shown the side and end elevation of a cornucopia which is to be constructed 
in eight pieces. The first step in the development of the pattern is a cor- 
rect representation of the article in these two views just named. It is not 
our purpose in this connection to describe in detail the method of drawing 
these two views. Certain parts must necessarily be conceived in the mind 
before they are laid upon the paper. For example, having determined that 
the article is to be constructed in eight pieces, and that its size at the mouth 
is to be of given dimensions, draw the section EFTBDCXA opposite 
its corresponding line, H 1 G', in the side elevation. Having determined the 
length of the article and its general shape, the profiles IP y 1 of the top, and 
G 1 c 1 of the bottom are draw^i. The end elevation is then worked out from 
these lines by means of corresponding lines carried across, after which the 
intermediate lines showing the side, which is turned directly toward the sight, 
are inserted, being derived in the same way from the sectional view. "We 
think this much of a general description will enable the intelligent reader to 
construct the necessary views of such an article. But, ordinarily, work of this 
character comes to the pattern cutter already drawn, the labor of delineating e 

it beinsj the work of a draftsman and designer, rather than that of the pat- Fig. 510.— lb Describe the Pattern of 
tern cutter. Accordingly, we commence our description with the assump- an 0ctagon shaft t0 m Agahlst a 

tion that the side and end elevations have been correctly drawn. By inspec- ^ ^ 












Fig 511. Patterns for the Volute of a Capital. 

tion of the side elevation, it will be seen that the profile of piece No. 1 is to be taken directly from the lower 



222 



Pattern Problems. 



1 L off t i r f? i T> ^ ^ angl6S t0 X °' f ° rmiD - the Side of tlie P lan «ing section No 

1, % off a stretchout taken from the profile G' c\ as shown by Q> C». Through the points used in lavL ff 

tins stretchout, draw measuring hues in the usual manner. From the corresponding points in the profile GV 

cany Imes across section No. 1 in the end elevation, cutting the two miter lines, as shown. With the T^are 




Fig. 512.- The Patterns for a Cornucopia in Eight Pieces. 



at right angles to the side X C, and brought successively against these points in the two miter line, cut corre- 

Td n c m i ™ r e g ti e r Th rv ines traced , tl r ish these severai "° iQts ° f -«-> - ss^is 

tteMe^ g TT V^ ° ? **?** °\ f" SeCti ° n " In ^ maMler take a streteW of ^e profile of 
ItSn 9 ? * i , S1 t 6 ^f 10n ' and lay U ° ff at ri ° ht aQ S les t0 tlie side of ^ plan F Y, bounding 

section No. 2, all as shown by W y\ through the points in which draw the usual measuring Hues. From the 



Pattern Problems. 22b 

points in the profile II 1 y l carry lines across the end elevation, cutting the miter lines hounding section No. 2. 
Then with the T-square at right angles to the side F Y of the plan, bringing it successively against the several 
points in the miter lines bounding section No. 2, cut the corresponding measuring lines. Then lines traced 
through these points of intersection, as shown by F'/ 1 and Y 2 y\ will be the pattern for section No. 2. For 
sections Nos. 3 and 4, situated diagonally to the section by which the stretchouts for the two patterns just 
described were obtained, another sectional view of the cornucopia must be constructed. To obtain stretchouts 
for these patterns, a section must be taken through the article at right angles to their respective sides. Our next 
step, therefore, is to construct a section corresponding to the line K L drawn through the plan, for which we 
proceed as follows : Through the point G 1 draw a horizontal line, G 1 g, upon which drop points from the profile 
G 1 o\ all as indicated by the small figures. At right angles to K L, at any point convenient for the required 
section, draw G 4 g\ in length equal to G 1 g, in which set off points corresponding to the points in G 1 g. Through 
these points draw lines after the usual manner of measuring lines. In order to obtain the points in the end 
elevation from which to draw lines cutting these measuring lines, by which to determine the diagonal section, 
we proceed as follows : From the points in the line D" d l carry lines cutting the lines A a in end elevation, 
and in like manner from the line B 2 I 1 carry lines cutting the line B I of the end elevation. By this means it 
will be seen that in the boundary lines of pieces Nos. 4 and 3 the same points have been obtained, both being 
derived from lines in the side elevation having corresponding divisions. Therefore, if these points be connected 
by drawing lines across the respective sections, and their middle points be taken, we shall have points of meas- 
urement by which to construct the diagonal section. The diagonal line K L cuts a number of these cross lines 
in the center, but the others, on account of the distortion of the end elevation, will fall at other points than on 
the line K L. It will be seen, for instance, that the line corresponding to S crosses section No. 4 ohliquely, but 
still its center point must be the center point in the section. Therefore, from the center point in the two sec- 
tions Nos. 4 and 3 thus determined, draw lines cutting the measuring lines drawn through G 4 g\ Then lines 
traced through these intersections, as shown by L 1 I and K' k, will form the diagonal sections of the article cor 
responding to a line, K L. For the pattern of No. 3 j>roceed as follows : At right angles to its side, TV B, lay 
off a stretchout corresponding to the side of the section constructed, agreeing with it. In other words, make 
the stretchout 1/ P equal to L 1 I of the section. Through the points draw measuring lines in the usual manner. 
Bring the T-square successively against the points in the miter lines bounding No. 3, placing the blade at right 
angles to the side Y B, and cut corresponding measuring lines. Then lines traced through the points of inter- 
section thus formed, as shown by B a V and Y 2 if, will form the pattern of No. 3. In like manner, for the pat- 
tern of No. 4 proceed as follows : At right angles to the side A X lay off a stretchout taken from the side of 
the section K 1 7c, which corresponds with it, all as indicated by K 7c 1 , through the points in which draw the 
usual measuring lines. Place the T-square at right angles to the sides A X, and, bringing it successively against 
the points in the miter lines bounding piece No. 4, cut the corresponding measuring lines. Then lines traced 
through the points of intersection thus obtained, as shown by X" of and A 1 a', will be the pattern for it. The 
pattern for No. 5 is obtained directly from the side elevation, as shown. That part of it in the smaller portion 
of the article is bounded by lines so nearly corresponding to the side elevation as to render it impossible in an 
engraving so small as here represented to distinguish between them. It begins to deviate, however, in a manner 
that may be shown in points corresponding to lino No. 4, and a description of this part will serve to illustrate 
the principle upon which the development of the pattern is based. Commencing with point 4, lay off a stretch- 
out taken from the corresponding portions of the profile G 1 C, as indicated by the small figures 3 2 1 in the 
line G 6 c\ Through these points draw measuring lines in the usual manner, and with the T-square placed at 
right angles to them, and brought successively against corresponding points in the sides of piece No. 5, as 
shown in the side elevation intersecting them, trace lines, all as indicated by the lines terminating at B' D 1 . 
Then B 1 V d' D 1 will be the pattern of piece No. 5. 

604. The Patterns for a SMjp Ventilator, having an Oval Mouth on a Round Pipe.— In Fig. 513 there 
are presented the front and side elevations of a style of ship ventilator occasionally employed. It starts from 
a round pipe, A 1 B\ at the base, and ends in an elliptical shape, as shown by O E P S, at the mouth. The rule 
which we present for developing the patterns is one allowing the mechanic the largest possible latitude in pro- 
portioning the article. It is also one which, with slight modifications, can be made to answer in the patterns of 
other ventilators of the same general kind which differ in the shape of the mouth. Care must he taken to 
draw the elliptical lines representing the sections, both in the elevation and in the development of the patterns, 
by the same means in all cases. For example, if a string and pencil or the trammels are used in drawing 



224 



Pattern Problems. 




Fig. 513.— Elevations and Section. 
A Ship Ventilator, having an Oval Mouth on a Round Pipe. 



KPSO, the same means should be used in drawing corresponding sections wherever they may he required. 
The reason for this is very simple. The principle upon which the rule is based is that an oblique section 

through a cylinder, and also a sec- 
tion through the opposite sides of a 
cone, is an ellipse. Having estab- 
lished the section through the article 
at either of the joint lines, both of 
the pieces which there meet must 
be based upon that section, so far 
as their stretchouts and other meas- 
urements are concerned, and there 
should be a correspondence between 
the several sections in this respect. 
To draw the section for the edge 
of one pattern piece with the tram- 
mels, and for the other which meets 
it from centers with the compasses, 
would hardly produce satisfactory 
results. It is believed that the 
method here j>resented, on account 
of its brevity, comparing it with 
other rules which might be used to 
accomplish the same result, is one 
that will be found of great service 
in practical work. If the patterns, 
as shown in Fig. 515, are not laid 
out very accurately and carefully, misfits will occur. By the very nature of the operation here described, slight 
variations in obtaining points of measurements will prove cumulative in character, each succeeding step leading 
further from the correct line. Hence the necessity of accuracy in applying the rule. Aside from the care 
necessary to be taken with the sections above mentioned, the parts may be proportioned according to the judg- 
ment of the designer and the requirements of the case. Let A 1 B 1 be the size of the pipe upon which the 
ventilator fits at the bottom. Let P and B S be the dimensions of the elliptical mouth. From these two 
sections proceed to draw the elevation A B D C. The lines AEGKMC and B F H L 2s" L> may be drawn 
at pleasure. Having determined their form, divide them by points, as shown by E6KM in the one and 
F H L X in the other, by which to locate the seams between the parts of which the article is to be composed. 
Connect these two sets of points by lines, as shown by E, F, G, H, etc. The lines B U and S V in the front 
elevation are to be drawn by eye rather than by any set rule. The only direction that needs to be given is to 
proportion their sweep to the width suggested by the outlines of the side. 
Their office in the development of the patterns is to determine the width of 
the several elliptical sections taken through the article. Therefore, if they are 
abrupt in their curve at any point, they are likely to produce an unsatisfactory 
outline in the finished work. By these two elevations the work is laid out as 
it is to be constructed. As will be evident from inspection of the engraving, 
a separate pattern will be required for each section. Since all of these, save 
the lower one, are alike in kind, though differing in size, a single example will 
be sufficient for showing the principles involved. The pattern of the section 
E A B F will be the same as that for the corresponding piece in an ordinary 
elbow, and may be developed as described in Section 511, and therefore need not be specially explained here. 
The patterns for the other sections will be developed as follows, taking M X D C as an example : This section, 
for convenience and in order to avoid confusion of lines, is transferred to the opposite side of the front elevation, 
as shown by "W T Z X. Bisect the several lines of seams between the sections. Thus, bisect the line C D, obtain- 
ing the point n. Bisect M X, obtaining the point m. In like manner locate h g e. These points are to be 
used in determining the width of the several elliptical sections, and for this purpose lines from them are carried 





Fig. 514 



C 5 

Diagrams of Triangles for 
Measurements. 

A Ship Ventilator, having an Oval 
Mouth on a Bound Pipe. 



Pattern Problems. 



29.* 



across the front elevation, cutting the lines K U and S V, as shown. Having drawn the section W Y Z X in 
line with the front elevation, as already described, drop points from Y and Z perpendicular to the section line 
O T of the elevation, thus locating the points M a and W. Make the distance Ir c equal to I c. Then draw 
the ellipse W V X 2 , which will be a plan or top view of the section M X of the side elevation. On a line par- 
allel with Y Z construct the section M? V W, as follows : Lot M 1 N' be equal to and opposite Y Z. Let the 
distance & V be equal to the distance c 1, of the section. With these points determined, draw the curve IT V W, 
which will be a regular ellipse. Divide the sections M' V JST" and S P into the same number of equal parts' 
as indicated by the small figures in the engraving. Drop the points 1, 2, 3, 4, etc., on to and perpendicular to 
the line Y Z ; thence carry them perpendicular to the center line O P of the front elevation, cutting the section 
W V X 2 in the points I s , 2 2 , 3 2 , etc., thus dividing it into the same number of spaces as were given to the origi- 
nal section M 1 V X". Xext connect the points of the same numbers in the two sections of the front elevation, 
thus: connect 2 1 with 2\ 3' with 3% 4' with i\ etc. ; also connect the points 2' with 1\ 3' with 2 2 , 4' with 3 2 , 
etc., all as shown in the engraving. These lines represent the bases of certain triangles, the vertical bights of 
which may be measured on the horizontal lines cutting the lines W X and W Z. The next step, therefore, is 
to construct diagrams of these triangles, as shown by A and B of Fig. 514. Draw any two horizontal lines as 
bases of the triangles, and erect the perpendiculars E C and F D. On both E C 
and F D set off the various bights of the triangles, measured as above stated and 
as indicated by the points 1, 2, 3, 4, etc. Xext set off the length of the bases of 
the triangles as follows : In diagram A, let C 1 equal the distance l 1 2 2 of Fig. 
1, make C 2 equal to 2 1 2 2 , make C 3 equal to 3 1 3", etc. Connect the points in 
the vertical lines with the points in the horizontal lines of the same number, thus 
obtaining hypothenuses of the triangles, or the true distance between the points 
l 1 1", 2 1 2 2 , etc., of the elevation. In diagram B, let the distances D 2, D 3, D 4, 
etc., represent the distances I 2 2 1 , 2" 3 1 , etc., of the elevation. Having located 
these points, connect 1 in the vertical line with 2 in the base ; also 2 in the verti- 
cal line with 3 in the base, and proceed in this manner for the other points. This 
will give the .hypothenuses of the triangles, whose bases are 1" 2 1 , 2" 3 1 , etc., 
in the elevation. Having thus obtained the true measurements of the vari- 
ous triangles in the envelope of the first section of the ventilator, proceed to 
develop the pattern for it, as shown in Fig. 515. On any straight line, C M, set 
off a distance equal to 1 1 in diagram A. From C as center, with radius equal 
to l 1 2 1 of the elevation, Fig. 513, draw an arc, which cut by another arc drawn 
from ~K as center, with radius equal to 1 2 of diagram B, thus establishing the 
point 2. From 2 as center, with radius equal to 2 2, diagram A, draw an arc, 
which cut with another arc drawn from V, Fig. 515, as center, with radius equal to 1 2 of the elevation, thus 
establishing the point 2'. Proceed in this manner, next locating the point 3, then the point 3 1 ; next the point 
4, and then 4', etc. It will be noticed that, after passing points 6 and 6 1 , T 1 is obtained before 7. Tins is 
for the sake of accuracy, as it will be seen by inspection of the elevation, Fig. 513, that the distance T C is 
less, and therefore more easily measured in the plan, than the distance from G 2 to 7'. Having thus located the 
points 1, 2, 3, etc., 1', 2", 3', etc., draw the lines C D and X M, as indicated in Fig. 515. Connect D and X. 
Then D X M C will be the pattern for one-half the section M X D C of the elevation. 

605. The Patterns for a Curved Tajjering Horn, Octagonal in Section.— Let a b e d I e f I in Fig. 51C 
represent a section of the article at the small end. Drop the points a I a d vertically to the horizontal line K P. 
"With any given radius, X Y, determined by the requirements of the case, and from a center upon the line K P. 
draw an arc, X Y, which will represent the center line through the article. From the point X in the line X P 
erect a vertical line, V G. Continue the center line horizontally beyond the vertical Y G, as shown by Y 1 '. 
Upon this line construct a section of the required article at the large end, as shown by A B C D W E F. 
From the points A B C D in this section carry points horizontally until they cut Y G, as shown by IITUG. 
Having thus located the points in the elevation at both large and small ends, complete the figure by drawing 
the arcs KG,JU,ZT and L H from center, which will be found in the line K P, all as indicated in the^ en- 
graving. Produce the line G Y indefinitely in the direction of H, upon which locate the points H 1 t u G' by 
duplicating points of corresponding letters in the upper part of the line derived from the larger section. Com- 
plete the plan view of the article by connecting the points it and a, t and I, IP and e, /and G', and c and I. 




515.— Pattern for First Section. 
A Ship Ventilator, having an Oval 
Mouth on a Hound Pipe. 



226 



Pattern Problems. 



Having thus constructed the several views of the article required for the patterns, proceed as follows : The 
side J U T Z may be pricked directly from the drawing. The pattern for the upper side, shown by K G in 

the elevation, may be obtained as follows : Upon any 
straight line, as K G in Fig. 517, lay off the stretchout 
of K G of Fig. 516, as indicated. Through the points 
K and G draw the perpendicular d k and D "W", mak- 
ing d k equal in length to d k of Fig. 516, and D "W 
equal in length to D ¥ of the same figure. Connect 
the points d D and k W, thus completing the pattern. 
The pattern for the lower side, shown by L H in ele- 
vation, is to be obtained in the same general manner, 
all as shown in Fig. 518. The pattern for the side 
ZTHL may be described as follows : Let the line 
a u of the plan view in Fig. 516 be considered the 
plane in which this face lies. From 0, which is the 
center of the inner arc of this piece, drop a line at 
right angles to L P, continuing it until it strikes the 
line of its plane a u, as shown by o. Thence carry 
it at right angles to the bine a u indefinitely in the 
direction of 11. Continue the line b a, which is the 
profile of this strip, until it intersects the line o B in 
the point E. Then B a will be the radius of the arc 
which will form the inner side of the pattern. In Fig. 
519, from B as center, with B a as radius, describe the 
arc a II, which in length make equal to the stretchout 
L II, Fig. 516, all as shown by the small figures. To 
obtain the line of the outer are of this piece, from the 
point in Fig. 516 draw a line to the point T, cutting 
the arc L II, as shown in the point 8. In transferring 
the stretchout this point S must be correctly located, 
as shown by S in the arc a LT, Fig. 519. From the 
center Pi, by which the arc a II was struck, draw a 
straight line through the point 8 indefinitely. Take 
the distance P> A in Fig. 516, which is the profile of 
the wide end of the required piece, as radius, and 
519, as center, describe an arc cutting B B in the point B. Draw I B, which will be the wide 




Fig. 516. — Elevation, Plan and Section. 
A Curved Tapering Horn, Octagonal in Section. 



From B, the center by which the inner arc of the pattern was struck, draw a straight line 



point a, producing it indefinitely in 



from H, Fig. 
end of the pattern, 
cutting the 

both directions. From a set off the distance a b, 
equal to a b of Fig. 516, which will be the width 
of the narrow end of the pattern. The only re- 
maining step necessary is to discover a radius, and a 
center in the line b B produced, by which an arc 
may be struck which will connect the points b and 
B. This, by experiment, will be found to be B'. 




517.— Pattern of Piece Corresponding to K G- of the Elevation. 
A Curved Tapering Horn, Octagonal in Section. 



-+r- V-+3— t— ^-ir-17-V " H 



For the pattern of the piece K 6 U J of Fig. 516 the ope- 
ration to be performed is very similar to that just described. From 
the point k in the side view draw a straight line to the point t, 
which consider the plane in which the outer arc of this piece lies. 
From the point P draw a line at right angles to K P, which produce 
until it intersects k t produced in the point p. Thence at right 
angles to k p draw the line p S indefinitely. Produce k e, which is 
the profile of the required piece at the narrow end, until it intersects the line last drawn in the point S. Then 
S k will be the radius of the arc which will form the outer line of the pattern. Transfer the line k e S to Fig. 



Fig. 51S. 



-Pattern for Piece Corresponding to L H of 
the Elevation. 



^1 Curved Tapering Horn, Octagonal in Section. 



Pattern Problems. 



227 



From the point 12 in this 




Fig. 519.— Pattern for Inside Flaring Piece. 

A Curved Tapering Horn, Octagonal in 

Section. 



520, as shown by d c S. From S as center, with radius S d, describe the arc d D, which in length make equal 

to the arc K G, Fig. 516. From the point P, Fig. 516, draw a line through the point U, cutting the arc K G 

in point 12. Carefully locate this point 12 in laying off the stretchout in^Fig. 520, 

figure draw a straight line to the center S, as shown. Take the distance 

D C of the large section, Fig. 516, between the feet of the dividers, and 

placing one foot on the point D in Fig. 520, swing the other foot around 

until it cuts the line S 12 in the point C. Then C D will be the wide 

end of the required pattern. Having now the two ends correctly laid off 

and the outer arc drawn, it remains to discover a radius, and a center in 

the line S c d, by which an arc may be struck connecting the points C. 

This is to be determined by experiment, from which it will be found that 

the center is S' and the radius S 1 c. This method is not to be considered 

mathematically correct. It is offered on account of its convenience for 

use and its close approximation to accuracy. It is believed it will be 

found of greater service in practical work than a rule in which principles 

are carried to an extreme, resulting in a long and tedious operation. 

606. In bringing this work to a close at this point, we do so not 
because the list of problems which might be presented has been exhaust- 
ed, but because we think enough has been given to serve every necessary 
purpose. New problems are Continually arising, and the number of com- 
binations which can be made between the various solids known to geom- 
etry, which alone can determine the pattern problems that might be enumerated, is almost infinite. The list to 
which we have given attention in the preceding pages has been gathered during the years in which The Metal 
Worker has been publishing articles upon pattern cutting, and accordingly is believed to embrace all of the 
more important problems arising in both tin-shop work and cornice making. The fact that many of the dem- 
onstrations were prepared in answer to questions propounded by correspondents of that journal, attests the prac- 
tical bearing upon ordinary workshop practice. In our selection of problems we have been disposed to give 

preference to those of an elementary character, and which 
are useful in work of almost daily occurrence, rather than to 
those of exceptional application, the demonstration of which 
could not be of interest to any considerable number of me- 
chanics. As elsewhere stated, our aim has been to state 
principles, with examples of their application, rather than 
to present arbitrary rules. Rules, when wanted, can be 
formulated to suit the pattern cutter's requirements, being 
based upon the principles which it has been our aim to 
explain. 

607. In almost every problem which occurs in practice 
the mechanic has the choice of several methods. Some- 
times these methods differ from each other only in minor 
particulars and are in reality the same. Still there is in 
many cases enough difference between them in this respect 
to warrant a choice. In other instances the difference be- 
tween methods is radical, making one much more advan- 
tageous for employment than the others. The careful pat- 
tern cutter will be on the lookout always for points of this 
kind. He will most carefully avoid falling into ruts or 
fixed habits, because in the ever-changing conditions of his 
work set methods often prove a disadvantage. Still other differences in ways of developing patterns may be 
noted in this connection. The available methods before the mechanic resolve themselves into two general 
classes. There are what are sometimes called shop rules, and mathematical rules. The former class, while 
including much that is of no practical value, contains some few methods which, on account of their brevity, as 
compared with mathematical rules, are really good. Shop rules in general are quite arbitrary in character, at 




Fig. 520.— Pattern for Outside Flaring Piece. 
A Curved Tapering Horn, Octagonal in Section. 



228 Pattern Problems. 

least upon first sight, but if there is anything in them of merit, upon closer examination an underlying mathe- 
matical principle will be found at the bottom of them. This brings us to say that many so-called shop rules 
may be devised by the intelligent pattern cutter which will be of great use and convenience. Many of the 
operations in pattern cutting, referring now to the usual mathematical rules, are simply routine in character, 
and when the mechanic has become sufficiently familiar with the results produced, some of them can be omit- 
ted, the net result being laid down arbitrarily by inspection. This is only a suggestion of a way by which shop 
rules can be devised. Intercourse with mechanics has shown that many of them prefer rules of this kind to 
those of a purely mathematical character. A few demonstrations properly belonging to this class will be found 
on the pages preceding, but the majority are mathematical. 



I N - D E X 



In order to make this -work of the greatest usefulness for occasional reference, the names of many articles of ware commonly made 
in tin shops, but which are not specifically mentioned in the pattern problems, have been incorporated in the index. The sections 
figures and pages given in connection with such articles, refer to rules which may be used in developing their patterns. 



Abbreviated method for cutting a square 

miter §427, p. 68 

Acute angle, An § 19, p. 2 

Acute-angled triangle § 3 r > P- 3 

Adjusting the drawing of a polygon to 
suit the requirements of miter cutting 

§ 309-373, P. 55 
Adjusting the edges of drawing tables. . 

§211, p. 16 
Advice to f tudents of pattern cutting, 

General §459, p. So 

Altitude §33, p. 3 

Altitude of a cylinder, The § 104, p. 8 

Altitude of a pyramid or cone, The. § 104, p 8 
Analysis of the solid to the envelope of 
which regular flaring ware corre- 
sponds § 455, p. 7S 

Angle, An §17, p. 2 

Angle, Elbow at any .§ 513, p. 124 

Angle into equal parts, To divide an. . . . 

§2S8, p. 32 

Angles § 17-20, p. 2 

Angl-s, Calculation of, by means of set 

squares § 339, p. 45 

Angles niaasure I by degrees. ..§ 72-73, p. 6 

Angular pediment, Au fig. 91, p. 12 

Antiquarian (drawing paper).. . .§ 274, p. 28 

Apex, The § 34, P- 3 

Approximate ellipse, To draw an, with 
the compasses to given dimensions, 
using three sets of centers. . . .§ 399, p. 61 
Approximate ellipse, To draw an, with 
the compasses to given dimensions, us- 
ing two sets of centers . . .§ 397-398, p. 60 
Approximate ellipse, To draw an, in a 
given rectangle by means of inter- 
secting Hues § 394, p. 60 

Approximate figure maybe constructed, 
To find tha centers iu a given ellipse 

by which an § 451, p. 61 

Arc § 63, p. 5 

Arc of a circle into any given number 

of equal parts, To divide an. .§ 302, p. 36 
Arc, The chord and bight of a segment 
being given, to find the center by 

which the, may be struck § 290, p. 33 

Arc, To find the center from which a 

given, is struck § 2S9, p. 33 

Arc, To find the center from which a 
given, is struck by the use of the 

square § 291, p. 33 

Architrave | 129, p. 10 

Area § 202, p. 14 

Arrangement of geometrical problems.. 

§ 276, p. 29 

Art and science of pattern cutting p. 63 

Atlas (drawing paper) § 274, p. 28 



218 



r8 



167 



Axis, An §111, p. 9 

Axis, Conjugate § 84, p. 7 

Axis, Transverse § 84, p. 6 

Axis, Envelope of a right cone from which 

a section is cut parallel to its. .§ 487, p. 99 
Axis, To find the centers and true, of an 

ellipse § 400, p. 61 

Ball, To construct a, in any number of 
pieces in the shape of gores. .§ 599, p. 
Ball, To construct a, in any number of 
pieces in the general shape of zones. . 
§ 598, p. 
Ball, To describe the patterns of an oc- 
tagon shaft to fib against a. . .§ 601, p. 
Ball, The pattern for a miter between 
the moldings of adjacent gables upon 
a square shaft formed by a.. .§ 555, p. 

Band, Dentil § 147, p. 11 

Band, MoJillion § 146, p. 11 

Base of a triangle § 35, P- 3 

Base is an oblong, The pattern of a flar- 
ing article of which the, and the top 

square § 471, p. 86 

Base is a true ellipse, The pattern of a 
flaring article which corresponds to 
the frustum of a coue whose. § 493, p. 103 
Base, The envelope of the frustum of a 
cone the, of which is an elliptical fig- 
ure § 492, p. 102 

Base, The pattern of an article having 

an elliptical, and a round top..§ 4S3, p. 96 
Base, The pattern of a flaring article 
the, of which is a rectangle and the 
top of which is round, the center of 
the top being toward one end..§ 4S1. p. 94 

Basin, Wash § 4SS, fig. 333 

Basis, A § 196, p. 13 

Bath, Hip §499, figs. 349-354 

Bath, Infant's § 485, fig. 329 

Bath, Plunge § 4S5, fig. 329 

Bath, Patterns for a hip §499, P- It0 

Bath, Sponge §488, fig. 333 

Bead capping the hip finish in a curved 
mansard roof, The patterns for the, 
the angli of the hip being a right an- 
gle §583, P- 204 

Beam compasses and trammels 

§ 240-243, p. 21 

Bed molding, A § 144. P- ll 

Bed molding at the top, Patterns for a 
hip niolJing upon a square mansard 

roof mitering pgainst a § 576, p. 

Bed molding of a deck cornice of a man- 
sard roof, Patterns for a hip molding 
mitering against the, which is square 
at base and octagonal at top.. g 5S1, p. 
Bed molding of corresponding profile, 



191 



199 



Patterns of a hip molding upon an oc- 
tagon angle in a mansard roof miter- 
ing against a § 578, p. 194 

Bed course § 145, p. n 

Bending metal to fit a given stay, A 

molding is made by § 407, p. ^4 

Bisect § 204, p. 14 

Blank for a curved melding. . . .§ 566, p. 177 
Blower for a grate, Pattern for.tj 534, p. 150 

Boards, Drawing § 212-215, P' 16 

Boiler, Coffee S 488, fig. 333 

Boiler, Flaring milk § 488, fig. 333 

Boss fitting over a miter in a molding, 

The patterns of a § 596, p. 216 

Bottom of which is oblong, The pattern 
of a flaring article the top of which is 
round and the, with semicircular ends, 
the top of the center being located 

near one end § 484, p. 97 

Bottom of which is oblong with semicir- 
cular ends, The pattern of a flaring 
article the top of which is round and 

the § 480, p. 93 

Bowl, Wash § 488, fig. 333 

Box of instruments, A § 269, p. 27 

Bracket § 134, p. 10 

Bracket in a curved pediment, A rak- 
ing § 573, P. 187 

Bracket head § 150, p. II 

Bracket melding § 150, p. n 

Bracket, The patterns for a drop upon 

the face of a § 595, p. 216 

Bracket, Patterns for a raking. § 572, p. 184 

Brackets and modillions compared 

§ 135, P- 10 

Brands of India ink fcj 264, p. 26 

Bread pan, Square § 468, fig. 295 

Broken pediment, A § 178, p. 12 

Broken pediment, In a, to ascertain the 
profile of the horizontal return at the 
top, together with its miters. S 55i> P- 163 

Bucket, Dutch §488, fig. 333 

Butt miter, A § 164, p. 12 

Butt miter against an irregular molded 

surface, A § 543, P- 156 

Butt miter against a plain surface shown 

in elevation, A §539, P- x 54 

Butt miter against a plain surface shown 

in plan, A §541, p. 155 

Butt miter against a regular curved sur- 
face, A § 540, P- 154 

Butt miter of a molding inclined iu ele- 
vation against a plain surface oblique 

in plan, A § 542, p. 155 

Calculation of angles by means of set 

squares § 339, P- 45 

Can, Kerosene § 486, figs. 330-331 



230 



Index. 



Cans, Oil § 4S6, figs. 33°-33i 

Cap § 274, p. 28 

Cap, The patterns in a common win- 
dow .§ 437, p. 70 

Capital, The volute of a § 602, p. 220 

Capping-, Kidge § 155, p. 11 

Carpenter's square, Construction of reg- 
ular polygons by the use of a p. 53 

Center by which the arc may be struck, 
The chord and hight of a segment of 
a circle being given, to find tbe.§2go,p. 33 
Center from which a given arc is struck, 

To find, by the use of square. .§ 291, p. 33 
Center from which a given arc is struck, 

To find the § 289, p. 33 

Center of a circle § 58, p. 4 

Center of the top being toward one end, 

The pattern of a flaring article the 

base of which is a rectangle and the 

top of which is round, the. . . .§ 4S1, p. 94 

Centers, To find the, and true axis of an 

ellipse § 400, p. 61 

Centers, To find the, in a given ellipse 
by which an approximate figure may 

be constructed § 4or, p. 61 

Central idea in miter cutting. . . .§ 423, p. 67 
Chamber pails § 488, fig. 333 



Chord. 



.8 66, 



P- 5 



Chord and hight of a segrnsnt of a cir- 
cle being given, to find the center by 
which the arc may be struck.. § 290, p. 33 

Circle § 56, p. 4 

Circle and an ellipse compared, Draw- 
ing a § 376, p- 55 

Circle, An oblique projection of a.§ 388, p. 57 

Circle, General rule for drawing a regu- 
lar polygon in a § 322, p. 41 

Circle, To divide an arc of a, into any 
given number of equal parts. .§ 302, p. 36 

Circle, To draw a, by a string and pen- 
cil § 376, p. 55 

Circle, To draw a, through three given 
points not in a straight line. . § 293, p. 34 

Circle, To draw a dodecagon within a 
given § 363, p. 52 

Circle, To draw an ellipse as the oblique 
projection of a § 389, p. 58 

Circle, To draw an equilateral triangle 
about a given § 351, p. 49 

Circle, To draw an equilateral triangle 
within a given. § 312, 343, 360, pp. 39, 46, 51 

Circle, To draw a hexagon about a 
given § 352, p. 49 

Circle, To draw a hexagon within a 
given § 345, p. 47 

Circle, To draw an octagon about a 
given § 353, p. 50 

Circle, To draw an octagon within a 
given § 346, 362, pp. 47, 52 

Circle, To draw a regular decagon within 
a given § 319, 321, pp. 40-41 

Circle, To draw a regular heptagon with- 
in a given § 316, p. 40 

Circle, To draw a regular hexagon wioh- 
in a given § 3T5, p. 39 

Circle, To draw a regular nonagon with- 
in a given § 318, p. 40 

Circle, To draw a regular octagon with- 
in a given § 317, p. 4° 

Circle, To draw a regular polygon of 
twelve sides within a given . . . § 323, p. 41 

Circle, To draw a regular undecagon 
within a given § 320, p, 40 

Circle, To draw a regular pentagon 
within a given § 314, p. 39 

Circle, To draw a square about a given. 

§ 354, P- 50 

Circle, To draw a square within a 
given § 313, 344, 361, pp. 39, 47, 52 

Circles and intersecting lines, To con- 
struct an ellipse to given dimensions 
by the use of t wo § 392, p. 5 7 

Circumference § 57, p. 4 



Circumference of a given circle, To draw 

a straight line equal to the 

§ 296-297, pp. 34-35 
Circumference of a given circle, To 
draw a straight line equal to the 

quarter § 299, p. 35 

Circumscribed polygons § 70, p. 5 

Class, Patterns of forms of the first. . . . 

§455, P- 63 
Classes, The forms with which the pat- 
tern cutter has to deal aie of two 

general § 454, p. 63 

Coal hod, Patterns of a § 500, p. 112 

Coffee boiler § 4S8, fig. 333 

Coffee filters, Flaring | 4S8, fig. 333 

Coffee pot, Flaring § 4SS, fig. 333 

Coffee pot, Octagonal. . . .§ 466, figs. 2S9-291 

Columbier (drawing paper) § 274, p. 28 

Companion, Housemaid's § 463, fig. 284 

Compasses and dividers. . . .§ 236-238, p. 20 
Compasses and straight edge in the con- 
struction of polygons p. 39 

Compasses, Beam, and trammels 

§ 2415-243, p. 21 
Compasses. To draw an approximate 
ellipse with the, to given dimensions, 
using two sets of centers. § 397-398, p. 60 
Compasses, To draw an approximate 
ellipse with the, to given dimensions, 
using three sets of centers. . . .§ 399, p. 61 
Compasses, To draw an elliptical figure 
with the, the length only being given 

§ 395, P- 60 
Complement of an arc or angle. . . § 74, p. 6 

Complete miter § 172, p. 12 

Composition of the shape of regular 

flaring ware § 458, p. So 

Concave, A § 123. p. 10 

Concentric circles § 63, p. 5 

Conclusion, A § 197, p. 13 

Cone, A § 97-ico, p. 8 

Cone, An irregular section through an. 

elliptical § 49S, p. 10S 

Cone, Development of the pattern for 

an elliptical § 452, p. 77 

Cone, Diagram of triangles used in the 
development of the pattern of an 

elliptical § 452, p. 77 

Cone, Frustum of a right § 450, p. 75 

Cone generated by the revolution of a 

right-angled triangle, A right. § 441, p. 72 
Cone intersecting a cylinder of greater 
diameter than itself at other than 
right angles, Frustum of a. . .§ 529, p. 141 
Cone, Method for describing the envel- 
ope of a right § 450, p. 75 

Cone, Method of developing the envel- 
ope of an elliptical § 451, p. 76 

Cone pattern and a miter pattern, Sim- 
ilarity between the measurements 
necessary to the development of 

a §444, P- 73 

Cone revolved to show the shape of its 

envelope, A. scalene § 445, p. 73 

Cone, Triangles for generating an ellip- 
tical § 452, p. 76 

Cone, To describe the shape of an ob- 
lique section of a, through its oppo- 
site sides § 391, p. 59 

Cone, To draw an ellipse as a section of 

a §391, P- 59 

Cone, The envelope of a frustum of a 

right § 4S8, p. 100 

Cone, The envelope of a frustum of a 
scalene, or the envelope of the section 
of a right cone contained between 

planes oblique to its axis § 491, p. 102 

Cone, The envelooe of a right. . .§ 486, p. 99 
Cone, The envelope of a right, from 
which a section is cut parallel to its 

axis § 4S7, p. 99 

Cone, The envelope of a scalene. § 490, p. 101 
Cone, The envelope of the frustum of 



a, the base of which is an elliptical 

figure § 492, p. 102 

Cone, The envelope of the frustum of a 
right, the upper plane of which is ob- 
lique to its axis § 489, p. 100 

Cone, The envelope of the section of a 
right, contained between planes ob- 
lique to its axis, or the envelope of the 
frustum of a scalene cone. . .§ 491, p. 102 
Cone, The patterns of a cylinder join- 
ing a, of greater diameter than itself 
at other than right angles.. . .§ 531, p. 144 ' 
Cone, The patterns of a cylinder or pipe 
and, meeting at right angles to their 

axes § 527, p. 138 

Cone, The patterns of a frustum of a, 
intersecting a cylinder, their axes be- 
ing at right angles §528, p. 140 

Cone, The patterns of the frustum of a. 
joining a cylinder of greater diameter 
than itself at other than right angles, 
the axis of the frustum passing to one 

side of axis of cylinder § 530, p. 143 

Cone, Properties of an elliptical. § 452, p. 76 
Cone, The revolution of a right, by 
which the shape of its envelope is 

described § 443, p. 73 

Cone whose base is a true ellipse, The 
pattern of a flaring article which cor- 
responds to the frustum of a.§ 493, p. 103 
Cones of unequal diameters, Ihe pat- 
terns of two frustums of, intersecting 
at other than right angles to their 

axes § 533, p. 14S 

Cones of unequal diameters, The patterns 
of two, intersecting at right angles to 

their axes § 532, p. 145 

Conic section, A § 117, p. g 

Conic sections § 390, p. 58 

Conical flange to fit around a pipe and 
against a roof of one inclination, A. . 

§507, p. 119 
Corneal shapes in pattern cutting. § 449, p. 75 
Conical spire mitering upon eight gables, 

The pattern of a § 563, p. 174 

Conical spire mitering upon four gables, 

The pattern of a § 560, p. 172 

Conjugate axis § 84, p. 7 

Construction of a trammel, The.§ 382, p. 56 
Construction of regular polygons by 

means of the protractor p. 51 

Construction of polygons by the use of 

compasses and straight-edge p. 39 

Construction of regular polygons by the 
use of the T-square, triangles, 0$ set- 
squares p. 45 

Construction of regular polygons by the 

use of a carpenter's square p. 53 

Construction of regular polygons, The. p. 39 
Construction of polygons, Table of divis- 
ions upon the square for the. .§ 368, p. 54 

Convex § 124, p. io 

Cooler, Flaring wine § 488, fig. 333 

Coping having a double wash, The pat- 
terns of a cylinder mitering over the 

peak of a gable § 505, p. 118 

Corbel, A § 142, p. n 

Corners, The patterns of a flaring arti- 
cle oblong in plan with rounded, and 
having a greater flare at the ends than 

at the sides § 477, p. 90 

Corners, The pattern of a regular flar- 
ing article with round § 474, p. 89 

Cornice § 126, p. 10 

Cornice at the corner of a building, A 
return miter at other than a right an- 
gle, as in a § 539, p. 153 

Corn'ce at the corner of a building, A 
square return miter or a miter at right 

angles, as in a § 538, p. 153 

Cornice, Deck § 137, p. 10 

Coi-nice, Lintel § 136, p. 10 

Cornice maker's drafting tables. § 209, p. 15 



Cornice of a mansard roof -which is 
square at the base and octagonal at 
the top, Patterns for a hip molding 
mitering against the bed molding f a 

deck § 5S1, p. igg 

Cornucopia, The patterns for a, in eight 

pieces § 603, p. 221 

Co-secant of an arc, The § S2, p. 6 

Co-sine of an arc, The g 80, p. 6 

Co-tangent of an arc, The g 7S, p. 6 

Course, Bed g 145, p. Ir 

Course, Dentil g 147, p. u 

Course, Modillion § 146, p. 11 

Cover, Wash boiler § 476, fig. 311 

Covering, A cylinder revolved, showing 

the shapa and extent of its. . .§ 447, p. 74 
Covering, A hexagonal pyramid revolved 
in such a manner as to describe the 

shape of its § 446, p. 74 

Covering for a molding § 410, p. 64 

Covering of a molding, A pattern for 

the. § 411-412, p. 65 

Covering for a molding, Use of lines in 

laying off the pattern of a. §412-413, p. 65 
Covering of a cube developed by revo- 
lution, The § 44S, p. 75 

Covering of a molding by means of a 
drawing, Obtaining the lines of meas- 
urement for the § 416-417, p. 66 

Covering of a triangular frustum ob- 
tained by revolution § 448, p. 75 

Cross joints, A pipe carried around a 

semicircle by means of § 514, p. 125 

Crown molding § 1 30, p. 10 

Cube, A §95, P- 7 

Cube, The covering of a, developed by 

revolution § 448, p. 75 

Cullender § 4SS, fig. 333 

Cup, Flaring § 4SS, fig. 333 

Cups for India ink § 267, p. 26 

Curve, A § 7, p. I 

Curved line, A § 8, p. I 

Curved molding, Blank for a. . .§ 566, p. 177 

Curved moldings. § 149, p. 11 

Curved moldings in a window cap, The 

patterns for elliptical § 569, p. 181 

Curved moldings in. a window cap, Pat- 
terns for simple § 568, p. 179 

Curved mansard roof. The pattern for 
tho bead capping the hip finish in a, 
the angle of the hip being a right an- 
gle § 5S3, P- 204 

Curved mansard roof, The pattern for a 
hip finish in a, the angle of the hip 

being-a right angle § 582, p. 202 

Curved mansard roof which is square at 
base and octagonal at top, Patterns 
of hip molding finishing a. . . .§ 584, p. 204 
Curved pediment, A raking bracket in a 

§ 573, P. 187 
Curved surface, A butt miter against a 

regular § 540, p. 154 

Curved tapering born, The patterns for 

a, octagonal in section § 605, p. 225 

Cutting a square miter, Abbreviated 

method for § 427, p. 6? 

Cutting a square miter, Short method 
and long method of. .§ 431-433. pp. 6S-69 

Cutting a square miter, Method of 

§ 427-435, PP- 68-70 
Cutting a square miter, The short rule 

for g4=7, P- 6S 

Cutting, Eule for avoiding errors in 

miter §435, P- 7° 

Cylinder, A . . . g 96, p. 7 

Cylinder (or pipe) and cone meeting at 
right angles to their axes, The pat- 
terns of a § 527, p. 138 

Cylinder generated by the revolution of 

the rectangle, A right g 441, p. 72 

Cylinder mitering over peak of a gable 

coping having double wash. .§ 505, p. 11S 
Cylinder of greater diameter than itself, 



Index. 

The frustum of a cone intersecting a, 

at other than right angles.. ..§ 529, p. 141 

Cylinder of greater diameter than itself, 
The patterns of the frustum of a cone 
joining a, at other than right augles, 
the axis of frustum passing to one side 
of the axis of the cylinder. .§ 530, p. 143 

Cylinder revolved, showing ihe shape in 
the patterns of an opening made in 

* t . ss l ide §447, p. 74 

Cylinder revo.ved, showing the shape 
and extent of its covering, A.§ 447, p. 74 

Cylinder, To describe the former shape 
of an oblique section of a . . . .§ 3Sg, p. 5S 

Cylinder, The patterns of a frustum of 
a coue intersecting a, their axes being 
at right augles g 528, p. 140 

Cylinder, The patterns of a, joining a 
cone of greater diameter than itself 
at other than right angles. . ..§ 531, p. 144 

Cylinder with one end cut obliquely and 
revolved so as to show the shape of its 
envelope, A §447, p. 74 

Decagon g 52, p. 4 

Decagon, The pattern for a newel post, 
the plan of which is a g 592, p. 212 

Decagon, To draw a regular, upon a 
given side § 331, p. 43 

Decagon, To draw a regular, within a 
given circle g 319, p. 40 

Deck cornice g 137, p. 1 [ 

Deck cornice of amansardroof, Patterns 
for a hip molding mitering against the 
bed molding of a, which is square at 
base and octagonal at top. . . .g 5S1, p. 199 

Deck cornice, The pattern of a hip mold- 
ing upon a right angle in a mansard 
roof mitering against the planceer of 

a : § 575, p. 189 

Deck corn ice upon a mansard roof which 
at the base is square and at the top 
octagon, Patterns of a hip molding 
mitering against the planceer of a. . . 

§ 5S0, p. iqS 

Deck molding g 137, p. n 

Definition of an ellipse 

§ 375, 379, 381, 3S7, 39°, PP. 55"5S 

Definition of miter g 414, p. 66 

Definitions and technicalities p. 1 

Degree, A g 72, p. 5 

Demonstration § 19:, p. 13 

Demy (drawing paper) g 224, p. 2S 

Dentil band g 147, p. n 

Dentil course g i-,7, p. n 

Dentil molding g 147, p. 11 



Detail drawing 



.818 



5, I>. 13 



Detail paper, Manila g 272, p. 27 

Develop a pattern. To § 198, p. 13 

Developing a molding in plastic material 

by means of a stay g 410, p. 64 

Developing the envelope of an elliptical 

cone §451, p. 76 

Development of a cone pattern and a 
miter pattern, Similarity between the 
measurements necessary to the. §444, p. 73 

Development of surfaces g 199, p. 13 

Development of the pattern for an ellip- 
tical cone g 452, p. 77 

Development of the pattern of an ellip- 
tical cone, Diagram of triangles used 

in the g 452, p. 77 

Diagonal g 55, P- 4 

Diagonal scales, Constructing . .g 249, p. 23 
Diagram of triangles used in the devel- 
opment of the pattern of an elliptical 

cone g 452, P- 77 

Diameter g 60, p. 4 

Diamond g 125, p. 10 

Diamond shape in plan, The envelope of 
the frustum of a pyramid which is. . 

§ 467, P- S4 
Dimensions by means of a trammel, To 
draw au ellipse to given g 384, p. 57 



59 



61 



P- 57 



231 

Dimensions, Drawing an ellipse to speci- 
fied, with a string and pencil. .§ 378, p. 56 
Dimensions, To construct an ellipse to ' ' 
given, by the use of two circles and 

intersecting lines g 392, p. 

Dimensions, To draw an approximate 

ellipse with the compasses to given 

using three sets of centers. . . § 399, p 

Dimensions, To draw an ellipse of given, 

by means of a square and a strip of 

wcod g 3 s6 

Di PPers g 4 S8, fiV 33^ 

Directrix g S5, p. 7 

Dish kettles 8 488, fig. 333 

Dish pans g 4 83, fig. 333 

Dishes for India ink .§ 267, p. 26 

Distances, Use of the T-square in trans- 
ferring ■■•••■■ -§439, P. 71 

Dividers, Compasses and. . .§ 236-238, p. 20 
Divisions of the protractor, The.g 357, p. 51 
Divisions upon the square for use in 

constructing polygons g 368, p. 54 

Dodecagon .8 53, p. 4 

Dodecagon, The pattern for an urn the 

plan of which is a § 593, p. 213 

Dodecagon, To draw a regular, upon a 

given side . § 333, p. 44 

Dodecagon, To draw a regu.ar, within 

a given circle § 321, p. 41 

Dodecagon within a given circle, To 

draw a g 363, p. 52 

Dodecahedron, The g 109, p. 8 

DoubleeIephant((lrawmgpaper).g 274, p~ 28 
Double wash, The patterns of a cylinder 
mitering for the peak of a gable cop- 
ing having a g 505, p. 118 

Draft, A g 183, p. 13 

Drafting tables g 208, p. 

Drawing, A g 184, p. 

Drawing a circle and an ellipse com- 
pared g 37 6, p. 

Drawing, A detail g 185, p. 

Drawing en ellipse with string and pen- 
cil. ...... .... §377, P- 

Drawing, A scale g 1S7, p. 

Drawing, A working g 186, p. 

Drawing boards g 212-215, p. 

Drawing boards, Testing g 217, p. 17 

Drawing of a polygon, Adjusting the, 
to suit the requirements of miter cut- 
ting § 309-373, P. 

Drawing paper in sheets g 274, p. 

Drawiug paper, White § 273, p. 

Drawing pens § 258-261, pp. 24-25 

Drawing pins g 268, p. 26 

Drawing tables, Adjusting the edges of. 

g 211, p. 16 
Drawing tables, Steel edges for.g 211, p. 16 

Drawing tables, Woods for g 210, p. 15 

Drawing tools and materials p. 15 

Dressing drawiug pens g 261, p. 25 

Drip § 133, p. 10 

Dripping pans g 46S, fig. 295 

Drop upon the face of a bracket, The 

patterns for a § 595, p. 216 

Dropping points, Use of the T-square in 

§ 439, P- 71 

Dutch buckets § 4SS, iig. 333 

Eccentric circles g 69, p. 5 

Edges of tables, Adjusting the. .g 211, p. 16 
Egg-shaped flaring pan, The pattern of 

an oval or g 478, p. 91 

Ecg-shaped figure, To draw an. .g 402, p. 62 

Elbow at any angle g 513, p. 124 

Elbow, A five-piece g 512, p. 123 

Elbow, A four-piece g 5 Ir , P- I22 

Elbow, A ihree'-piece g 5'°, P- '21 

Elbow, A two-pi ce g 509, p. 121 

Elbow, A two-piece, in elliptical pipo. .. 

§ 520, p. 133 
Elbow in flaring pipe, A three-piece, the 

middle section of which is straight. . . 

§ 5i8, p. 130 



15 
13 

55 
13 

56 
13 
13 
16 



55 
28 
27 



232 



Index. 



Elbow in tapering pipe, A two-piece . . . 

§516, p. 127 
Elbow in tapering pipe, A three-piece. . 

§ 517, P- 129 

Elbow, the middle section of which 
tapers, Three-piece § 519, p. 131 

Elephant (drawing paper) § 274, p. 28 

Elevation, An § 179, p. 12 

Ellipse §84, 118, pp. 6, 9 

Ellipse, A definition of the 

§ 375. 379, 38i, 3S7- 39°, PP- 55"5S 

Ellipse, A familiar talk about. . .§ 374, p. 55 

Ellipse compared, Drawing a circle and 
an §376. P- 55 

Ellipse, Drawing an, to specified dimen- 
sions with a string and pencil. § 378, p. 56 

Ellipse, Drawing an, with string and 
pencil § 377, p. 56 

Ellpse, The p. 55 

Ellipse, To construct an, to given dimen- 
sions by the use of two circles and in- 
tersecting lines § 392, p. 59 

Ellipse, To draw an approximate, in a 
given rectangle by means of intersect- 
ing lines § 394, p. 60 

Ellipse, To draw an approximate, with 
the compasses to given dimensions, 
using three sets of centers. . . .§ 399, p. 61 

Ellipse, To draw an approximate, with 
the compasses to given dimensions, 
using two sets of centers. § 397-398, p. 60 

Ellipse, To draw aD, as a section of a 
cone § 391 , p. 59 

Ellipse, To draw an, as the oblique pro- 
jection of a circle § 3S9, p. 58 

Ellipse, To draw an, of given dimensions 
by means of a square and a strip of 
wood § 386, p. 57 

Ellipse, To draw an, within a given rec- 
tangle by intersecting lines. . .§ 393, p. 59 

Ellipse, To draw an, to given dimensions 
by means of a trammel § 384, p. 57 

Ellipse, To find the centers and true axis 
of an § 400, p. 61 

Ellipse, To find the centers in a given, 
by which an approximate figure may 
be constructed § 401, p. 61 

Ellipse, The pattern of a flaring article 
which corresponds to the frustum of a 
cone whose base is a true. . . .§ 493, p. 103 

Elliptical at the base and round at the 
top, The pattern of an irregular flar- 
ing article which is, the top being so 
situated with respect to the base as to 
be tangent to one end of it when 
viewed in plan § 496, p. 107 

Elliptical base, The pattern of an article 
having an, and a round top. . .§ 483, p. 96 

Elliptical cone, An irregular section 
through an § 498, p. 108 

Elliptical cone, Diagram of triangles 
used in the development of the pat- 
tern of an § 452, p. 77 

Elliptical cone, Development of the pat- 
tern for an § 452, p. 77 

Elliptical cons, Method of developing 
the envelope of an § 451, p. 76 

Elliptical cone, Triangles for generating 
an § 452, p. 76 

Elliptical cone, The properties of an . . . 

§ 452, p. 76 

Elliptical curved moldings in a window 
cap, The patterns for § 569, p. 181 

Elliptical figure, To draw an, with the 
compasses, the length only being giv- 
en § 395, P- 60 

Elliptical figure, The envelope of the 
frustum of a cone the base of w hich 
is an § 492, p. 102 

Elliptical frustum is a part, The solid of 
which a regular flaring § 458, p. 80 

Elliptical pipe, A two-piece elbow in . . . 

§ 520, p. 133 



Elliptical pipe, Pattern of an, to fit against 
a roof of one inclination S302, p. 116 

Elliptical vase, The patterns for an, 
constructed in twelve pieces. § 594, p. 214 

Emperor (drawing paper) § 274, p. 2S 

End of an oblong pan, Pattern of the 
flaring, when the top is curved and 
the bottom is straight § 536, p. 151 

End of an oblong pan, Pattern of tne 
flaring, when both bottom and top of 
the flaring end are curved. . .§ 535, p. 150 

End upright, Tne pattern of a rectangu- 
lar flaring article having one..§ 470, p. 85 

Ends, The pattern of a regular flaring 
article which in shape is oblong with 
semicircular § 475, p. 89 

Euds, The pattern of a flaring article, 
the top of which is round and the bot- 
tom of which is oblong with semicir- 
cular § 480, p. 93 

Ends, Pattern of a raised cover fitting 
an oblong vessel with round. .§ 476, p. 90 

Entablature § 127, p. 10 

Envelope, An § 112, p. 9 

Envelope, A cylinder with one end cut 
obliquely and revolved so as to show 
the shape of its § 447, p. 74 

Envelope, A scalene cone revolved so as 
to show the shape of its § 445, p. 73 

Envelope is described, The revolution of 
a right cone by which the shape of its 

§ 443, P- 73 

Envelope of an elliptical cone, Method 
of developing the § 451, p. 76 

Envelope of a frustum of a right cone, 
The §488, p. 100 

Envelope of a frustum of a scalene cone, 
or the envelope of the section of a 
right cone contained between planes 
oblique to its axis, The § 491, p. 102 

Envelope of the frustum of a pyramid 
which is diamond shape in plan, The. 

§ 467, P- 84 

Envelope of a hexagonal pyramid, The. 

§ 464, P- 82 

Envelope of a right cone, The. .§ 486, p. 99 

Envelope of a right cone, Method for 
describing the § 450, p. 75 

Envelope of a right cone from which a 
section is cut parallel to its axn, The. 

§ 487, P- 99 

Envelope of a scalene cone, The. §490, p 101 

Envelope of a square pyramid, .§ 462, p. 81 

Envelope of which regular flaring ware 
corresponds, Analyses of the solid to 
the § 455, p. J8 

Envelope of the frustum of a cone, the 
base of which is an elliptical figure, 
The § 492, p. 102 

Envelope of the frustum of a square 
pyramid, The § 463, p. 82 

Envelope of the frusturnof aright cone, 
the upper plane of which is oblique to 
its axis, The §489, p. 100 

Envelope of the frustum of an octagonal 
pyramid having alternate long and 
short sides, The § 466, p. 83 

Envelope of the frustum of an octagonal 
pyramid, The § 465, p. 82 

Envelope of the solid, The string or 
thread method for developing the, to 
which regular flaring ware corre- 
sponds § 457, p. 79 

Envelope of a triangular pyramid, The. 

§461, p. 81 

Envelope of which regular flaring ware 
corresponds, The solid to the. .§ 453, p. 77 

Equal flare throughout, The patterns of 
a tapering article with § 494, p. 104 

Equilateral triangle § 27, p. 2 

Equilateral triangle al out a given cir- 
cle. To draw an § 351, p. 49 

Equilateral triangles, To construct a hex- 



agon by means of six, the length of a 
side being given § 367, p. 54 

Equilateral triangle, To construct an, 
length of a side being given. .§ 365, p. 53 

Equilateral triangle, Patterns for a ped- 
estal of which the plan is an.§ 5S5, p. 207 

Equilateral triangle upon a given side, 
To draw an § 347, p. 48 

Equilateral triangle upon a given side, 
To construct an § 324, p. 4) 

Equilateral triangle within a given cir- 
cle, To draw an 

§ 312, 343, 360, PP- 39, 46. 5' 
Errors in miter cutting, Rule for avoid- 
ing § 435, P. 7C 

Evolute, An § 87, p. 7 

Evolutent, An § 88, p. 7 

Extent of its covering, A cylinder re- 
volved showing the shape and.§ 447, p. 74 
Face of a bracket, The patterns for a 

drop upon the § 595, p. 216 

' Face of a molding § 157, p. 12 

I Face or panel miter, A square. . .§ 434, p. 6c 
Face, Patterns for a keystone with sink 

I in § 571, P- 181 

Face miters § 437, p. 7c 

Face miter or miter at right angles, as 

in a molding around a panel. § 546, p. 158 
Familiartalk about the ellipse, A.§ 374, p. 55 
Fastening paper to drawing boards. ... 1 

§ 216, p. I? 

' Figure, A plane § 23, p. 2 

Figure may be constructed, To find the 
centers in a given ellipse by which an 

approximate § 401, p. 61 

Figure, To draw an egg-shaped. .§ 402, p. 62 
Figure, To draw an elliptical, with the 
compasses, the length only being 

given § 395, p. 60 

Figures, Eectalinear § 24, p, 2 

Fillers, Lamp § 488, fig. 333 

Fillet, A § 173, p. 12 

Filters, Flaring coffee. § 488, fig. 333 

Finish are developed, The principles up- 
on which the plane surfacts of man- 
sard § 574, p. 189 

Finish in a curved mansard roof, The 
pattern for the bead capping the bip, 
the angle of the hip being a right an- 
gle § 583, P- 204 

Finish, The pattern for a hip, in a curved 
mansard roof, the angle of the hip be- 
ing a right angle. § 582, p. 202 

Five-piece elbow, A § 512, p. 125 

Flange, A § 174, p. 12 

Flange, A conical, to fit around a pipe 
and against a roof of one inclination 

§ 507, P. II? 
Flange to fit around a pipe and over the 

ridge of a roof, Pattern of a.§ 508, p. 12" 
Flange, The pattern of a, to fit around 
the pipe and against the roof of one 

inclination § 506, p. lie 

Flare throughout, The patterns of a ta- 
pering article with equal. . . .§ 494, p. loc 
Flaring article having one end upright, 

The pattern of a rectangular. .§ 470, p. 3* 
Flaring article, The pattern of a, which 
corresponds to the frustum of a cone 
whose base is a true ellipse. .§ 493, p. 10 j 
Flaring article, The pattern for an irreg- 
ular, which is elliptical at the base and 
round at the top, the top being so sit- 
uated with respect to the base as to be 
tangent to one end of it when viewed 

in plan § 496, p. I0 k " 

Flaring article, The pattern of a regular, 
which in shape is oblong with semicir- 
cular ends § 475, p. Sg 

Flaring article, The patterns of a, ob- 
long in plan with rounded corners, 
and having greater flare at the ends 
than at the sides § 477, p. gB 



Flaring article, The pattern of a, the 
base of which is a rectangle and the 
top cf which is round, the center of 
the top being toward one end.§ 4S1, p. 94 
Flaring article, The pattern of "an ob- 
long, having a round top. . . . .§ 4S2, p. 95 
Flaring article, The pattern of an irreg- 
ular, both top and bottom of which 
are round, the top being smaller than 
the bottom, and the two being tan- 
gent at one point in plan §495, p. 106 

Flaring article, The pattern of a, the 
top of which is round and the bottom 
of which is oblong with semicircular 
ends, the center of the top being 

located near one end § 484, p. 97 

Flaring article, The pattern of a, of 
which the base is an oblong and the 

top square § 471, p. 86 

Flaring article, Pattern of a, square at 

the base and round at the top.§ 473, p. 87 
Flaring article, The pattern of a rectan- 
gular § 468, p. 84 

Flaring article, The pattern of a, the top 
of which is round and the bottom of 
which is oblong with semicircular 

ends § 4S0, p. 93 

Flaring coffee filters § 4SS, fig. 333 

Flaring coffee pot § 4SS, fig. 333 

Flaring cups. § 488, fig. 333 

Flaring elliptical frustum is a part, The 

solid of which a regular § 458. p. 80 

Flaring end of an oblong pan, Pattern 

of tue, when both bottom and top of 

the flaring end are curved. . .§ 535, p. 150 

Flaring end of an oblong pan, Pattern 

of the, when the top is curved and 

bottom is straight § 536, p. 151 

Flaring measures § 488, fig. 333 

Flaring o'olong article, The pattern of a 

regular, with round corners. .§ 474, p. 89 
Flaring pan, The pattern of an oval or 

egg-shaped § 478, p. 91 

Flaring pans, Oval ......§ 492, figs. 337-339 

Flaring pans, Pound § 4SS, fig:. 333 

Flaring pipe, Three-piece elbow in, middle 
section of which is straight. .§ 513, p. 130 

Flaring tea kettles "...'§ 4SS, fig. 333 

Flaring tea pot § 488, fig. 333 

Flaring tray, The pattern of a heart- 
shaped § 479, p. 92 

Flaring ware corresponds, Analyses of 
the solid to the envelope of which reg- 
ular. .. . § 455, P- 78 

Flaring ware, Composition of the shape 

of regular § 458, p. 80 

Flaring ware corresponds, The solid to 

the envelopeof which regular. § 453, p. 77 
Flaring ware corresponds, The string 
method for developing the envelope 
of the solid to which regular. .§ 457, p. 79 
Flaring ware, The usual method for de- 
scribing patterns of regular. . .§ 454, p. 77 

Flaring wine cooler § 488, fig. 333 

Flat scales § 251-252, p. 23 

Float, A soap maker's § 537, p. 151 

Foci § 84, 86, pp. 6-7 

Focus § 85, p. 7 

Foot molding § 148, p. 11 

Foot tubs. Oval. ....... .§ 492, figs. 337-339 

Form or surface generated by a profile, 

A molding described as a § 40S, p. 64 

Forms, A molding in a succession of 

parallel § 407, p. 64 

Forms of first class, Patterns of .55 405, p. 63 
Forms with which the pattern cutter 
has to deal are of two general classes, 

The .§ 404, p. 63 

Four-piece elbow, A § 511, p. 122 

Four sided figure, A molding mitering 

around an irregular § 547, p. 159 

Frieze § 128, p. 10 

Frieze panel § 128, p. 10 



Index. 

Frieze piece § 128, p. 10 

From a given horizontal molding, to es- 
tablish the profile of a corresponding 
inclined molding, to miter with it at 
right angles in plan, and the several 

miter patterns involved § 550, p. 162 

From a. given horizontal profile, to estab- 
lish the profile for a corresponding in- 
clined molding, to miter with it at an 
octagon angle in plan, and the several 

miter patterns involved § 552, p. 165 

From a given profile in an inclined 
molding, to establish the profile of a 
corresponding horizontal molding, to 
miter with it at an octagon angle in 
plan, and the miter patterns. § 553, p. 166 

Frustum is a part, Tne solid of which a 
regular flaring elliptical § 45S, p. 80 

Frustum of a cone § 115, p. g 

Frustum of a eone, A § 100, p. 8 

Frustum of a cone, The patterns of the, 
joining a cylinder of greater diameter 
than itself at other than right angles, 
the axis of the frustum passing to one 
side of the axis of cylinder. .§ 530, p. 143 

Frustum of a cone the base of which is 
an elliptical figure, The envelope of 
the § 492, p. 102 

Frustum of a cone, The patterns of a, 
intersecting a cylinder, their axes be- 
ing at right angles § 52S, p. 140 

Frustum of a cone whose base is a true 
ellipse, The pattern of a flaring arti- 
cle which corresponds to the.g 493, p. 103 

Frustum of a cone intersecting a cylin- 
der of greater diameter than itself at 
other than right angles, The.§ 529, p. 141 

Frustum of an octagonal pyramid hav- 
ing alternate long and short sides, The 
envelope of the § 466, p. 83 

Frustum of an octagonal pyramid, The 
envelope of the § 465, p. 82 

Frustum of a pyramid which is diamond 
shape in plan, Envelope of §467, p. 84 

Frustum of a pyramid. . . § 115, p. 9 

Frustum of a right con» § 450, p. 75 

Frustum of a right cone, The envelope 
of a. § 488, p. 100 

Frustum of a right cone, the upper plane 
of which is oblique to its axis, The en- 
velope of the § 489, p. 100 

Frustum of a scalane cone, The envelope 
of a, or the envelope of the section of 
a right cone contained between planes 
oblique to its axis § 491, p. 102 

Frustum of a solid g 115, p. 9 

Frustum of a square pyramid, The en- 
velope of the '. § 463, p. 82 

Frustum obtained by revolution, Cover- 
ing of a triangular .'. . .§ 443, p. 75 

Frustums of cones, The patterns of two, 
of unequal diameters intersecting at 
other than right angles §533, P- 148 

Finial, The patterns of a, the plan of 
which is octagon with alternate long 
and short sides § 591, p. 210 

Funnels § 486, figs. 330-331 

Gable, A § 165, 190, pp. 12-13 

Gable coping having a dcuble wash, The 
patternsof a cylinder mitering 1 or the 
peak of a § 505, p. 118 

Gable molding, A § 154, P- n 

Gable miters, Simple § 54S, p. 160 

Gables, Miter between the moldings of 
adjacent, upon a square shaft, the ga- 
bles being of different pitches. § 556, p. 168 

Gables of an octagon pinnacle, Patterns 
for the moldings and roof pieces in 
the § 557, P- 170 

Gables of a square pinnacle, Patterns 
for the moldings and roof pieces in 
the § 554,. P- 166 

Gables, Miter between moldings of adja- 



233 

cent, upon an octagon shaft, the ga- 
^ bles beiusr of different pitches. § 55S, p. 170 
Gables, The pattern of an octagon spire 

miterins upon four .§ 561, p. 173 

Gables, The pattern of a square spire 

mitering upon four g 559, p, i^i 

Gables, The pattern of an octagon spire 

mitering upon eight .§ 562, p. 174 

Gables, The pattern of a conical spire 

mitering upon eight § 563, p. 174 

Gables, The pattern of a conical spire 

mitering upon four g 560, p. 172 

Gables upon a square shaft formed by 

means of a ball, The pattern for a 

miter between the moldings of adja- 

rt Cent V J --. §"5. f- 167 

General advice to students of pattern 

cutting § 4,9, p . 80 

General rule for pattern cutting. § 426, p. 6S 
General rule for drawing any regular 

polygon in a circle § 322, p. 4r 

General rule for drawing any regular 

polygon, the length of a sido being 

„ S iveu § 334, p. 44 

b-eometry g 2, p. I 

Geometrical problems .p. 29 

Given dimensions by means of a tram- 
mel, To draw an ellipse to. . . .8 384, p. 57 
Given dimensions, To construct an el- 
lipse to, by the use of two circles and 

intersecting lines g 392, p. 59 

Given dimensions, To draw an approxi- 
mate ellipse with the compasses to, 
using three sets of centers. . . .§ 399, p. 61 
Give a dimensions, To draw an ellipse 
of, by means of a square and a strip 

of wood g 386, p. 57 

Given dimensions, To draw an approxi- 
mate ellipse wilh the compasses to, 
using two sets of centers. g 397-398, p. 60 
Given ellipse, To find the centers in a, 
by which an approximate figuro may 

be constructed § 401, p. 61 

Given line, A .§ g, p. 1 

Given point, A g 9, p. 1 

Given rectangle, To draw an approxi- 
mate ellipse in a, by means of inter- 
secting lines g 394, p. 60 

Given rectangle, To draw an ellipse 
within a, by means of intersecting 

lines §393, P- 59 

Globe, A g 101, p. 8 

Gore piece forming transition between 

an octagon and a square g 565, p. 176 

Gore piece in a molding forming transi- 
tion from an octagon to a square. 

The § 564, p. 175 

Gores, A ball in any number of pieces 

in the general shape of g 599, p. 2i3 

Grate, Pattern for a blower for.g534, p. 150 

Grades of India ink g 264, p. 25 

Grecian ogee, To draw a g 334, p. 36 

Grocer's scoops g 5og, fig. 369 

Guide for striking the segment of a cir- 
cle, Triangular § 2g2, p. 34 

Hair-spring dividers § 238, p. 20 

Head, Bracket g 150, p. 11 

Head block, A g 141, p. 11 

Heart shaped flaring tray, The pattern 

of a g 479. V- 9 2 

Heptagon g 49, P- 4 

Heptagon, To draw a regular, upon a 

given side § 3 2S , P- 4 2 

Heptagon, To draw a regular, within a 

given circle g 316, p. 40 

Heptagon, The patterns for a vase tho 

plan of which is a § 5S9, p. 209 

Hexagon g 48, P- 4 

Hexagon about a given circle, To draw 

a §352, p. 49 

Hexagon, To construct a, by means of 
six equilateral triangles, the length of 
a side being given g 367, p. 54 



234 



Index. 



Hexagon, To construct a, the length of 

a side being given § 366, p. 53 

Hexagon, To draw a regular, upon a 

given side § 327, 349. PP- 42, 48 

Hexagon, To draw a regular, within a 

given circle §315, 345, PP- 39, 47 

Hexagon, The pattern for a pedestal, 

the plan of which is a § 5S8, p. 20S 

Hexagonal prism, A § 94, p. 7 

Hexagonal pyramid ..--■§ 102, p. S 

Hexagonal pyramid revolved in such a 
manner as to describe the shape of its 

covering, A § 44°, P- 74 

Hexagonal pyramid, The envelope of a. 

§ 464, p. S2 

Hexahedron, A § 107, p. 8 

Hip, A .§189, p. 13 

Hip bath § 499, fi g s - 349"354 

Hip bath, Patterns for a § 499,^ p. no 

Hip finish in a curved mansard root, The 
pattern for the bead capping the, the 
angle of the hip being a right angle. . 

§ 5S3, p. 204 
Hip finish, The pattern fcr a, in a 
curved mansard roof, the angle of 
the hip being a right angle. .§ 5S2, p. 202 

Hip molding, A § 156, P- " 

Hip molding mitering against the bed 
molding of a deck cornice of a man- 
sard rouf which is square at the base 
and octagonal at the top, Patterns for 

a § 581, p. 199 

Hip molding 011a mansard roof, Patterns 
for the miters at the bottom of a, 
which is octagon at the top and square 

at the bottom § 579, P- 195 

Hip molding, Patterns for a, mitering 
against the planceer of a deck cornice 
upon a mansard roof which at the base 

is square and at the top octagon 

§ 580, p. 198 
Hip molding, Patterns of a, finishing a 
curved mansard roof which is square 
at the base and octagonal at the top. . 

§ 5S4, p. 204 
Hip molding upon an octagon angle in a 
mansard roof mitering against a bed 
molding of a corresponding profile, 

Patterns of a § 578, p. 194 

Hip molding upon an octagon angle of a 
mansard roof mitering against an in- 
clined wash at the bottom, The pat- 
terns of a § 577, P- T 93 

Hip molding upon a right angle in a man- 
sard roof mitering against tbe plan- 
ceer cf a deck cornice, The pattern 

of a § 575, P- 189 

Hip molding upon a square mansard roof 
mitering against bed molding at the 

top, Patterns for a § 576, p. 191 

Hod, Patterns of a coal g.500, p. 112 

Home-made set squares § 235, p. 20 

Horizontal line in a drawing, Method of 

indicating a g 12, p. 2 

Horizontal lines g n, p. 2 

Horizontal molding, A § 151, p. n 

Horizontal molding, From a gi\ en pro- 
file in an inclined molding to establish 
the profile of a corresponding, to miter 
with it at an octagon angle in plan, 

and the miter patterns involved 

§ 553, P- 166 
Horizontal molding, To ascertain the pro- 
file of a, adapted to a miter with a 
given inclined molding at right angles 
in plan and the several miter patterns 

involved § 549. "p. 160 

Horizontal molding, to establish the pro- 
file of a corresponding inclined mold- 
ing, From a given, to miter with it at 
right angles in plan, and the several 

miter patterns involved § 550, p. 162 

Horizontal profile, From a given, to es- 



tablish the profile for a corresponding 
inclined molding, to miter with it at 
an octagon angle in plan, and the seve- 
ral miter patterns involved. .§ 552, p. 165 
Horizontal return at the top, together 
with its miters, In a broken pediment 
to ascertain the profile of the.g 551, p. 163 

Horizontal section § 181, p. 12 

Horn, The patterns for a curved taper- 
ing, octagonal in section § 605, p. 225 

Housemaid's companion § 463, fig. 284 

Hyperbola § 86, 120, pp. 7, 9 

Hyperbola and parabola compared 

§ 121, p. 10 

Hypothenuse, A § 33, p. 3 

Hypothesis § 193, p. 13 

Icosahedron, The § no, p. 9 

Idea in miter cutting, Central. ..§ 423, p. 67 
Identity of long and short rule for cut- 
ting square miters g 438, p. 78 

Imperial (drawing paper) § 274, p. 28 

Improvised trammel, An § 385, p. 57 

In a broken pediment, to ascertain the 
profile of the horizontal returns at 

the top, with its miters § 551, p. 163 

Incised work .§ 1S8, p. 13 

Inclinedlines § 15, p. 2 

Inclined molding, An § 153, p. 11 

Inclined molding at right angles in plan 
and the several miter patterns in- 
volved, To ascertain the profile of a 
horizontal molding adapted to a miter 

with a given § 549, p. 160 

Inclined molding, From a given profile 
in an, to establish the profile of a cor- 
responding horizontal molding, to 
miter with it at an octagon angle in 

plan, and the patterns involved 

§ 553, P- 166 
Inclined molding, From a given horizon- 
tal profile to establish the profile for a 
corresponding, to miter with it at an 
octagon angle in plan, and the several 

miter patterns involved g 552, p. 165 

Inclined molding, To establish the pro- 
file of a corresponding, from a given 
horizontal molding, to miter with it at 
right angles in plan, and the several 

miter patterns involved § 550, p. 162 

Inclined wash at the bottom, The pat- 
terns of a hip molding upon an octa- 
gon angle of a mansard roof iriter- 

ing against an § 577, p. 193 

Indefinitely § 206, p. 14 

India ink. ". § 262-26S, pp. 25-26 

India rubber § 270-271, p. 27 

Infant's bath. , § 485, fig. 329 

Ink, India § 262-26S, pp. 25-26 

Inscribed circles § 71, p. 5 

Inscribed polygons § 70, p. 5 

Inside miter § 169, p. 12 

Instruments, A box of § 269, p. 27 

Intersecting lines, To construct an el- 
lipse to given dimensions by the use 

of two circles and § 392, p. 59 

Intersecting lines, To draw an ellipse 
within a given rectangle by means of 

§ 393. P- 59 
Intersecting lines, To draw an approxi- 
mate ellipse in a given rectangle by 

means of § 394, p. 60 

Intersection of solids .§ 114, p. 9 

Involute, An § 8S, p. 7 

Ionic volute, To describe an. . . .§ 307, p. 37 
Irregular flaring article, The pattern 
for an, which is elliptical at the base, 
round at the top, the top being so sit- 
uated with respect to the base as to be 
tangent to one end of it when viewed 

in plan § 496, p. 107 

Irregular flaring article, The pattern of 
an, both top and bottom of which are 
round, the top being smaller than the 



bottom and the two being tangent at 

one point in plan § 495, p. 106 

Irregular four-sided figure, Patterns of a 

molding mitering around an..§ 547, p. 159 
Irregular section through an elliptical 

cone, An .§ 49S, p. 108 

Irregular surface, A molding terminat- 
ing against an § 427, p. 67 

Isosceles triangle § 28, p. 3 

Jars, Slop § 48S, fig. 333 

Joint at other than right angles between 
two pipes of different diameters, the 
axis of the smaller being placed to one 
side of that of the larger. ...§ 526, p. 138 
Joint between two pipes of different di- 
ameters intersecting at other than 

right angles § 525, p. 137 

Joint between two pipes of same diame- 
ter at other than right angles.. §524, p. 136 

Kerosene cans § 486, figs. 330-331 

Kettles, Dish § 4SS, fig. 333 

Kettles, Milk (hood) § 4S9, fig. 334 

Kettles, Wash § 4S8, fig. 333 

Keystone, The patterns for tho face and 

side of a plain tapering § 570, p. 182 

Keystone with sink in face, Patterns 

for a § 571, p. 182 

Lamp fillers § 48S, fig. 333 

Lead pencils § 253-257, p. 24 

Line, A § 5, p. 1 

Line, A curved § 8 p. I 

Line, A miter, not usually employed in 

cutting a square miter § 427, p. 68 

Line, A given § 9, p. 1 

Line, A straight § 6, p. 1 

Line equal to the circumference of a 

given circle, To draw a straight 

§ 296-297, pp. 34-35 
Line equal to the semi-circumference of 
a given circle, To draw a straight. . .. 

§ 298, p. 35 
Line equal to the quarter circumference 
of a given circle, To draw a straight. 

§ 299, p. 35 
Line, To divide a given straight, into 



two equal parts. . 



53-2S5. p. 31 



Line into any number of equal parts, To 

divide a given straight.§2S6-287,pp. 31-32 
Line, Mistake in placing miter. .§ 434, p. 69 
Line parallel to another line, To draw a 

§ 278, p. 30 

Lines, Measuring § 413, p. 66 

Lines, Stretchout g 413, p. 66 

Lines, Horizontal § II, p. 2 

Lines, Inclined § 15, p. 2 

Lines, Parallel. . .• § 10, p. 2 

Lilies, Perpendicular § 16, p. 2 

Lines of measurement for the covering 
of molding by means of a drawing, 

Obtaining the § 416-417, p. 66 

Lines, To construct an ellipse to given 
dimensions by the use of two circles 

and intersecting § 392, p. 59 

Lines, To draw an approximate ellipse 
in a given rectangle by means of in- 
tersecting g 394, p. 60 

Lines, To draw an ellipse within a given 
rectangle by means of intersecting. . 

§ 393, P- 59 
Lines Use of, in laying off the pattern 
of a covering for a molding 

§ 412-413, P- 65 

Lines, Vertical § 13, p. 2 

Lintel cornice § 136, p. 11 

Lipped measures (lip) § 533, fig. 401 

Lipped measures (body) g 488, fig. 333 

Long and short rule for cutting square 

miters, Identity of § 43S, p. 70 

Long method of cutting a square miter 

Short compared and.. g 431-433, pp. 6S-69 
Longitudinal seams, To form a semicir- 
cle in a pipe by means of. . . .§ 515, p. 126 
Lozenge '. . .§ 43, p. 3 



Manila detail paper § 272, p. 

Mansard finish are developed, The priii- ' 
ciples upon which the plain surfaces 



= 7 



of. 



,, , ••;•• 8 574, p. 1S9 

Mansard roof, Pattern for a hip molding 
upon a square, mitering against bed 
molding at the top § 576, p. IgI 

Mansard roof, Patterns for a hip molding 
mitering against the bed molding of a 
deck cornice of a, which is square at 
base and octagonal at top. . . .§ sSr, p. 

Mansard roof, Patterns of a hip molding 
upon an octagon angle in a, mitering 
against a bed molding of correspond- 
ing proftlo § 57 8 ? p . 

Mansard roof, Patterns for the miters at 
the bottom of a hip molding on a, 
which is octagon at the top and square 
at the bottom g c^g p 

Mansard roof, The patterns or a hip 
molding upon an octagon angle of a, 
mitering against an inclined wash at 
the bottom g 57^ p. 

Mansard roof, The pattern of a hip- 
molding upon a right angle in a, mi- 
tering against the planceer of a deck 

cornice g jyj ) p 

Mansard roof, Pattern for bead capping 
the hip finish in a curved, the angle of 
the hip being a right angle.. . .§ 583, p 
Mansard roof, The pattern for a hip fin- 
ish in a curved, the angle of the hip 



199 



194 



195 



193 



1S9 



.204 



being a right angle. 



333 

64 
15 



66 



Mansard roof which is square at base and 
octagonal at top, Pattern of a hip 
molding finishing a curved. ..§ 5S4, p. 

Marking pots §488, fi°-. 

Material by means of a stay, Developing 

a molding in plastic § 410, p. 

Materials, Drawing tools and p. 

Measurement for the covering of mold- 
ing by means of a string, Obtaining 

the fines of § 4x6-417, p. 

Measurements necessary for the devel- 
opment of cone and miter patterns, 

Similarity between the § 444, p. 73 

Measurements upon surfaces, Miter cut- 
ting is a system of g 436. p. 70 

Measures, Flaring g 4S8, fig. 333 

Measures, Lipped (lip) g 533, fig. 401 

Measures, Lipped (body) g 488, fig. 333 

Measuring lines . .g 413, p . 66 

Medium (drawing paper) g 274, p. 28 

Metal, A molding is made by bending, 

to fit a given stay g 407, p. 64 

Method for cutting a square miter, Ab- 
breviated § 427, p. 6S 

Method for describing the envelope of a 

right, cone g 450, p. 75 

Method for describing patterns of reg- 
ular flaring ware, The usual . .§ 454, p, 77 
Method for developing the envelope of 
the solid to which regular flaring ware 

corresponds, The string g 457, p. 7g 

Method of cutting a square miter 

§ 427-435, PP- 68-70 
Method of developing the envelope of an 

elliptical cone g 451, p. 76 

Method of indicating a horizontal line 

in a drawing g 12, p. 2 

Method of indicating vertical lines in a 

drawing g 14, p. 2 

Methods, Short, and long for cutting a 
square miter compared. §431-433, pp. 68-69 

Milk boilers, Flaring § 488, fig. 333 

Milk kettles (hood) § 489, fig. 334 

Milk pans g 488, fig. 333 

Milk strainer pails (body) § 488, fig. 333 

Milk strainer pails (hood) g 489, fig. 335 

Milk strainers g 488, fig. 333 

Minutes g 72, p. 5 

Mistake in placing miter line. . . .§ 434, p. 6g 
Miter g 163-172, p. 12 



Index. 

Miter against an irregular molded sur- 
face, A butt g 543, p . I5 6 

Miter against a plain surface shown in 
elevation, A butt g 5 3 g, p. 1^ 

Miter against a plain surface shown in 
plan, A butt g 54Ii p . I55 

Miter against a curved surface, A butt. 

Miter, Abbreviated method for outl rag a 
square g 427i p . 63 

Miter, A miter line not usually employed 
in cutting a square g 427, p. 68 

Miter at other than a right angle, A re- 
turn, as in a cornice at the corner of 
a building g 53g> p . I53 

Miter at right angles, A square return 
miter or a, as in a cornice at the cor- 
ner of a building g 538, p. I53 

Miter, A square face or panel. .8 434, p. 69 

Miter, A square return g 428, p. 68 

Miter between the moldings of adjacent 
gables upon a square shaft, the gables 
being of different pitches. . . .g 556, p. 168 

Miter between moldings of adjacent ga- 
bles upon an octagon shaft, the gables 
being of different pitches §-558, p. 170 

Miter between two moldings of different 
profiles g 544? p . I50 

Miter compared, Short and long methods 
of cutting a square. .§ 431-433, pp. 68-69 

66 



235 



Molding, A ridge § 155, p. 

Molding and roof pieces in gables of 
an octagon pinnacle, Patterns for 



Molding, 
Molding, 
Molding, 



. .. , §,557, p. 170 
A vertical § 152, p . ri 

§ 144. P- 11 



Bed. 



Miter cutting g 41 

Miter cutting, Adjusting the drawing of 

■8 582, p. 202 a polygon to suit g 369-373, p. 55 

Miter cutting is a system of measure- 
ments upon surfaces g 436, p. 70 

204 Miter cutting, Rule for avoiding^ errors 

m g 435, p. 70 

Miter cutting, The T-square in. .g 418, p. 66 

Miter, Definition of g 414, p. 66 

Miter, Face, or miter at right angles, as 

in a molding around a panel. g 546, p. 158 
Miter in a molding, The patterns of a 
boss_ fitting over a g 596, p. 216 



..Bracket S 150, p. 11 

Molding by means of a string, Obtain- 
ing the lines of measurement for the 

covering of a g 416-417, p. 66 

Molding, Covering for a g 410, p. 64 

Molding, Deck g I37; p . „ 

Molding described as a form or surface 

generated by a profile, A g 408, p 64 

Molding, Dentil.. .....§ 147, p. n 

Molding, Developing a, in plastic mate- 
rial by means of a stay g 410, p. 64 

Molding, Foot g 148, p . r [ 

Molding, From a given horizontaf, to es- 
tablish the profile of a corresponding 
inclined molding, to miter with it at 
right angles in plan, and the several 

miter patterns involved g 550, p. 

Molding, From a given horizontal pro- 
file to establish the profile of a corre- 
sponding inclined, to miter with it at 
an octagon angle in plan, and the sev- 
eral miter patterns involved. § 552, p. 
Molding, From a given profile in an in- 
clined, to establish the profile of a cor- 
responding horizontal molding, to 
miter with it at an octagon angle in 
plan, and the miter patterns involved. 

166 



162 



165 



64 



Miter lines, Mistake in placing. .§ 434, p. 69 

Miter line not usually employed in cut- 
ting a square miter, A § 427, p. 6S 

Miter, Method of cutting a square 

g 427-435, pp. 68-70 

Miter of a molding inclined in elevation 
against a plain surface oblique in plan, 

, r Abm : fc §542, p. 155 

Miter piece g 171, p. 12 

Miter, Similarity between the measure- 
ments necessary to the development 

of c, cone pattern and a § 444, p. 73 

Miter, The patterns for a, between the 
moldings of adjacent gables upon a 
equar; shaft formed by means of a 

ball g 555, p. 167 

Miter, The short rule for cutting a square 

§427, p. 68 
Miter, The usual method of developing a 

square return g 429, p. 6S 

Miters, Faee g 437, p. 70 

Miters, Identity of long and short rule 
for cutting square g 438, p 



195 



191 



. 70 
Miters, Patterns of simple gable. g 548, p. 160 

11 
1 1 
11 
10 



Modillion band g 146, p 

Modillion course g 146, p 

Modillion molding g 146, p. 

Modillions g 135, p. 

Modillions and brackets compared 

§135, P- 10 

Molding, A g 143, p. 11 

Molding around a panel, Face miter or 
miter at right angles, as in a.g 546, p. 158 

Molding, A gable g 154, p. 11 

Molding, A hip g 156, p. 11 

Molding, A horizontal g 151, p. 11 

Molding, An inclined g 153, p. 11 

Molding, A pattern for the covering of a 

§411-412, p. 65 
Molding, A raked g 160, p. 12 



8 553, p. 

Molding is a succession of parallel forms, 
-^ § 407, p 

Molding inclined in elevation, A butt 
miter of a, against a plain surface 
oblique in plan g 542, p. 155 

Molding is made by bending metal to fit 
a given stay, A g 407, p. 64 

Molding, Modillion g 146, p. n 

Molding on a mansard roof, Patterns for 
the miter at tho bottom of a hip, 
which is octagon at the top and 
square at the bottom g 579, p. 

Molding, Pattern for hip, upon a square 
mansard roof mitering against bed 
molding at the top g 576, p. 

Molding, Patterns of a hip, upon an oc- 
tagon angle in a mansard roof miter- 
ing agaiust a bed molding of a corre- 
sponding profile § 578, p. 194 

Molding, Patterns for a hip, mitering 
against the planceer of a deck cornice 
upon a mansard roof which at the 
base is square and at the top octa- 
gon § 5S0, p 

Molding, Patterns of a hip, finishing a 
curved mansard roof which is square 
at basa and octagonal at top.g 5S4, p 

Molding terminating against an irregu- 
lar surface, A § 421, p. 67 

Molding, To ascertain the profile of a 
horizontal, adapted to miter with a 
given inclined molding at right angles 
in plan, and the several miter patterns 

involved g 549, p. 

Molding, The blank for a curved, g 566, p. 

Molding, The gore piece in a, forming 

transition from an octagon to a 

square. ... g 564, p. 

Molding, The patterns of a, mitering 
around an irregular four-sided figure. 

§ 547, P- 159 
Molding, The pattern of a hip, upon a 
right angle in a mansard roof miter- 
ing against the planceer of a de--k 

cornice § 575, p, 

Molding, The patterns of a boss fitting 

over a miter in a § 596, p. 

Molding, The patterns of a hip, upon an 
octagon angle of a mansard roof mi 



19S 



204 



160 
177 



175 



189 
216 



236 



Index. 



tering against an inclined wash at the 

bottom § 577, p. 193 

Molding, To rake a § 159, p. 12 

Molding, Use of lines in laying off the 

pattern of covering for a § 412-413, p. 65 
Moldings and roof pieces under the ga- 
bles of square pinnacles. Patterns 

for the --§554, P- 166 

Moldings, Curved § 149, p. 11 

Moldings of adjacent gables upon an oc- 
tagon shaft, the gables being of dif- 
ferent pitches, The miter between 

the § 558. P- 170 

Moldings of adjacent gables upon a 
square shaft, Miter between the, the 

gables being of different pitches 

§ 556, p. 16S 
Moldings of different profiles, Miter be- 
tween two § 544, P- 156 

Moldings in a window cap, The patterns 

for elliptical curved § 5^9, p. 181 

Moldings in a window cap, Patterns for 

simple curved § 568, p. 179 

Moldines, Rake § 159, p. 12 

Moldings, The patterns for a miter be- 
tween the, of adjacent gables upon a 
square shaft formed by means of a 

bail § 555, P- 167 

Moldings upon a panel, the shape of 
which is a scilene triangle, The pat- 
terns of the § 545, P- 158 

Molds, J'udding § 488, fig. 333 . 

Names of set squares § 340, p. 46 

Newel post, The pattern for a, the plan 

of which is a decagon § 592, p. 212 

Nonagon § 51, p. 4 

Nonagon, To draw a regular, within a 

given circle § 318, p. 40 

Nonagon upon a given side, To draw a 

regular § 330, p. 43 

Normal profile, The § 162, p. 12 

Normal stay, The § 162, p. 12 

Oblique cone, An § 99, p. 8 

Oblique project ion of a circle, An. § 388, p. 57 
Oblique projection of a circle, To draw 

an ellipse as the § 389, p. 58 

Oblique section of a cone through its 
opposite sides, To describe the shape 

of an § 39T, p. 59 

Oblique section of a cylinder, To de- 
scribe the form or shape of an .8 389, p. 58 

Oblique sections § 181, p. 12 

Oblong article, The pattprnof a regular 

flaring, with round corners. . .§ 474, p. 89 
Oblong flaring article, The pattern of an, 

having a round top § 482, p. 95 

Oblong in plan with rounded corners, 
and having a greater flare at the ends 
than at the sides, The pattern of a 

flaring article § 477. p. 90 

Oblong pan, Pattern of the flaring end 

of an, when both botlom and top of 

the flaring eud are curved . . .§ 535, p. 150 

Oblong pan, Pattern of the flaring end 

of an, when the top is curved and 

bottom is straight § 536, p. 151 

Oblong, The pattern of a regular flar- 
ing article which in shape is, with 

semicircular ends § 475, p. 89 

Oblong tapering article with -one end 
square and one end semicircular, hav- 
ing more flare at the ends than at 
the sides, The patterns of an..§ 485, p. 98 
Oblong vessel with round ends, Pattern 

of a raised cover fitting an. . .§ 476, p. go 
Obtaining the lines of measurement for 
the covering of molding by means of 

a string §416-417, p. 66 

Obtuse angle. An § 20, p. 2 

Obtuse-angled triangle § 32, p. 3 

Octagon § 50, p. 4 

Octagon about a given circle, To draw 
an §353, V- 5° 



Octagon and a square, The gore piece 

forming transition bet ween an. §565, p. 176 
Octagon angle of a mansard roof mi- 
tering against an inclined wash at the 
bottom, The patterns of a hip mold- 
ing upon an § 577, p. 193 

Octagon angle in a mansard roof, Pat- 
terns of a hip molding upon an, mi- 
tering against a bed molding of cor- 
responding profile § 578, p. 194 

Octagon at the top and square at the 
bottom, Patterns for the miters at the 
bottom of a hip molding on a mansard 

roof which is § 579, p. 195 

Octagon coffee pot § 466, figs. 289-291 

Octagon miter § 168, p. 12 

Octagon pinnacle, Patterns for the mold- 
ings and roof pieces in the gables of 

an § 557, p. 170 

Octagon shaft fitting over the ridge of a 

roof, The pattern of an § 503, p. 117 

Octagon spire mitering upon eight ga- 
bles, The pattern of an § 562, p. 174 

Octagon shaft to fit against a ball, To 

describe the pattern of an. . .§ 601, p. 220 
Octagon shaft, The miter between the 
moldings of adjacent gables upon an, 
the gables being of different pitches.. 

§ 558, p. 170 
Octagon spire, The pattern of an, mi- 
tering upon four gables § 561, p. 173 

Octagon to a square, The gore piece in 
a molding forming transition from an 

§ 564, P. 175 
Octagon, To draw a regular, within a 

given square § 336, p. 45 

Octagon, To draw a regular, upon a 

given side § 329, p. 43 

Octagon, To draw a regular, within a 

given circle ..., §317, p. 40 

Octagon, The patterns of a finial the 
plan of which is, with alternate long 

and short sides § 591, p. 210 

Octagon teapot § 466, figs. 289-291 

Octagon upon a given side, To draw 

an §350, p. 48 

Octagon within a given circle, To draw 

an § 346, 362, pp. 47, 52 

Octagon pedestal, The pattern for an. . . 

§ 590, p. 209 
Octagonal at the top, The patterns of a 
tapering article which is square at the 

base and § 472, p. 87 

Octagonal pyramid having long and 
short sides, The envelope of the frus- 
tum of an § 466, p. 83 

Octagonal pyramid, The envelope of 

the frustum of an § 465, p. 82 

Octagonal shaft, The patterns of an, the 
profile of which is curved, mitering 

upon the ridge of a roof § 597, p. 217 

Octagonal vase, Patterns of an..§ 439, p. 71 

Octahedron, The § 108, p. 8 

Ogee, A Grecian § 304, p. 36 

Ogee drawn by means of quarter cir- 
cles § 303, p. 36 

Oil cans § 4S6, figs. 330-331 

Opening made in its side, A cylinder re- 
volved showing the shape in the pat- 
tern of an § 447, p. 74 

Outside miter § 170, p. 12 

Outside stay § 409, p. 64 

Oval figure, To draw an egg-shaped or. 

§ 402, p. 62 

Oval flaring pans §492, figs. 337~339 

Oval foot tubs § 492, figs. 337-339 

Oval or egg-shaped flaring pan. The pat- 
tern of an § 478, p. 91 

Oyster stew pans § 488, fig. 333 

Pails, Chamber § 48S, fig. 333 

Pails, Milk strainer (body). . . .§ 488, fig. 333 
Pails, Milk strainer (bood). . . .§ 489, fig. 334 
Pan, Pattern of the flaring end of an 



oblong, when the top is curved and 

bottom is straight § 536, p.. 151 

Pan, Pattern of the flaring end of an 
oblong, when both bottom and top of 
the flaring end are curved. . .§ 535, p. 150 
Pan, The pattern of an oval or egg- 
shaped flaring § 478, p. 91 

Panel, Face miter or miter at right an- 
gles, as in a molding around a. § 546, p. 158 
Panel miter, A square face or. . .§ 434, p. 69 
Panel, The patterns of the moldings 
upon a, the shape of which is a sca- 
lene triangle § f 45, p. 158 

Pans, Dish 8 4S8, fig. 333 

Pans, Dripping § 468, fig. 295 

Pans, Oval flaring § 492, figs. 337-339 

Pans, Milk § 4S8, fig. 333 

Pans, Square § 468, fig. 295 

Pans, Square bread § 468, fig. 295 

Paper § 272, p. 27 

Paper, Fastening, to drawing boards. . . 

§ 216, p. 17 

Parabola ,. . .§ 85, 119, pp. 7, 9 

Parabola and hyperbola compared 

§ 121, p. 10 
Parabola by the intersection of lines, its 
bight and ordinate being given, To 

draw a § 305, p. 36 

Parallel forms, A molding is a succession 

of § 407, P- 64 

ParalK 1 lines § 10, p. 2 

Parallelogram § 42, p. 3 

Pattern cutter has to deal are of two 
general classes, The forms with which 

the §4°4, P- 63 

Pattern cutting, A treatise on § 1, p. I 

Pattern cutting, Conical shapes in 

§ 449- P- 75 
Pattern cutting, General advice to stu- 
dents of § 459, p. 80 

Pattern cutting, General rule for.§ 426, p. 68 
Pattern cutting illustrated, The use of 

the T-square in § 439, p. 71 

Pattern cutting, Requirements for 

§425, P- 68 

Pattern cutting, Eules in § 459, p. 80 

Pattern cutting, Sheet metal § 3, p. 1 

Pattern cutting, Art and science of . . .p. 63 
Pattern for an elliptical cone, Develop- 
ment of the § 452, p. 77 

Pattern for a hip finish in a curved man- 
sard roof, the angle of the hip being 

a right angle, The § 582, p. 202 

Pattern for an irregular flaring article 
which is elliptical at the base, round 
at the top, the top being so situated 
with respect to the base as to be tan- 
gent to one end of it when viewed in 

plan, The § 496, p. 107 

Pattern for a miter between the mold- 
ings of adjacent gables upon a square 
shaft formed by means of a ball, The 

§555, P- 167 
Pattern for a newel post, the plan of 

which is a decagon § 592, p. 212 

Pattern for octagonal pedestal. .§ 590, p. 209 
Pattern for a pedestal the plan o£ which 

is a hexagon, The § 588, p. 208 

Pattern for a pedestal square in plan, 

The § 586, p. 207 

Pattern for a scale scoop, The. .§ 497, p. 107 
Pattern for an urn the plan of which is 

a dodecagon, The § 593, p. 213 

Pattern for the covering of a molding, 

A § 411-412, p. 65 

Pattern of an aiticle having an ellipti- 
cal base and a round top, The. 8, 483, p. 96 
Pattern of a conical spire mitering upon 

four gables, The § 560, p. 172 

Pattern cf a conical spire miterir.g upon 

eight gables, The § 563, p. 174 

Pattern of a covering of a molding, Use 

of lines in laying off the. .§412-413, p. 65 



Index. 



237 



Pattern c f an elliptical pipe to fit against 
a roof of one inclination § 502, p. 116 

Pattern of a flange to fit around a pipe 
and against a roof of one inclination, 
The § 506, p. 119 

Pattern of flange to fit around a pipe and 
over the ridge of a roof, The.§ 50S, p. 120 

Pattern of a flaring article which corre- 
sponds to the frustum of a cone whose 
base is a true ellipse, The. . . .§ 493, p. 103 

Pattern of a flaring article square at the 
base and round at top, The. . .§ 473, p. 87 

Pattern of a flaring article, the base of 
which is a rectangle and the top of 
which is round, the center of the top 
being toward one end, The. . .§ 48 r, p. 94 

Pattern of a flaring article of which the 
base is an oblong and the top square. 

§ 477, P- 86 

Pattern of a flaring article, the top of 
which is round and the bottom of 
which is oblong with semicircular 
ends, The § 4S0, p. 93 

Pattern of a heart-shaped flaring tray, 
The § 479. P- 9 2 

Pattern of an irregular flaring article, 
both top and bottom of which are 
round, the top being smaller than the 
bottom, and the two being tangent at 
one point in plan, The § 495, p. 106 

Pattern of a flaring article, the top of 
which is round anil the bottom of 
which is oblong with semicircular 
ends, the center of the top being 
located near one end, The. . . .§ 484, p. 97 

Pattern of an oblong flaring article hav- 
ing a round top, The § 4S2, p. 95 

Pattern of an octagon shaft fitting over 
the ridge of a roof, The § 503, p. 117 

Pattern of an octagou shaft to fit against 
a ball, To describe the § 601, p. 220 

Pattern of an octagon spire miteriug 
upon eight gables, The § 562, p. 174 

Pattern of an octagon spire mitering 
upon four gables, The § 561, p. 173 

Pattern of an opening made in its side, 
A cylinder revolved showing the shape 
in the §447. P- 74 

Pattern of an oval or egg-shaped flaring 
pan, The §478, p. 91 

Pattern of a laised cover fitting an ob- 
long vessel with round ends. .§ 476, p. 90 

Pattern of a rectangular article, three 
sides of which are vertical, the fourth 
being inclined, The § 469, p. 85 

Pattern of a rectangular flaring arti- 
cle, The § 468, p. 84 

Pattern of a rectangular fining article 
having one end upright, The . .§ 470, p. 85 

Pattern of a regular flaring article, 
which in shape is oblong with semi- 
circular ends § 475. P- 89 

Pattern of the flaring end of an oblong 
pan when the top is curved and bot- 
tom is straight § 53°, p. I5 1 

Pattern of the flaring end of an oblong 
pan when both bottom and top of the 
flaring end are curved § 535, p. 150 

Pattern of a regular flaring oblong arti- 
cle with round corners, The. .§ 474. p. 89 

Pattern of a round pipe to fit over the 
ridge of a roof § 5°4, P- "7 

Pattern of a round pipe to fit against 
roof of one inclination, The.§ 501, p. 116 

Pattern of a square spire mitering upon 
four gables, The § 559. P- J 7i 

Pattern problems p. 81 

Pattern, Similarity between the meas- 
urements necessary to development 
of cone pattern and miter § 444, p. 73 

Patternforablowerforagrate.g 534,. P- 150 

Patterns, A plain window cap and its 
several § 5°7, P- 178 



Patterns for a cornucopia in eight 
pieces § 603, p. 22 r 

Patterns for a curved tapering horn 
octagonal in section, The. . . .§ 605, p. 225 

Patterns for a drop upon the face of a 
bracket. The § 595, p. 216 

Patterns for an elliptical vase constructed 
in twelve pieces, The § 594, p. 214 

Patterns for a hipbath § 499, p. no 

Patterns for a hip molding upon a square 
mansard roof mitering against bed 
molding at the top § 576, p. 191 

Patterns for a hip molding mitering 
against the bed molding of a deck 
cornice of a mansard roof which is 
square at the base and octagonal at 
the top § 5S1, p. 199 

Patterns for a hip molding mitering 
against the planceer of a deck cor- 
nice upon a mansard roof, which at 
the base is square and at the top octa- 
gon § 580, p. 198 

Patterns for a keystone with sink iu 
face g 571, p. 182 

Patterns for a pedestal, of which the 
plan is an equilateral triangle. The. . . 

§ 585, p. 207 

Patterns for a raking bracket. .§ 572, p. 184 

Patterns for a ship ventilator having an 
oval mouth on a round pipe..§ 605, p. 223 

Patterns for a soapmaker's float. § 537, p. 151 

Patterns for a vase or urn in any num- 
ber of pieces. The § 440, p. 72 

Patterns for a vase the plan of which 
is a pentagon, The § 587, p. 207 

Patterns for a vase the plan of which is 
a heptagon § 5S9, p. 209 

Patterns for the bead capping the hip 
finish in a curved mansard roof, the 
angle of the hip being a right antrle, 
The § 583, p. 204 

Patterns for the face and side of a plain 
tapering keystone, The § 570, p. 182 

Patterns for the miters at the bottom of 
a hip molding on a mansard roof which 
is octagon at the top and square at the 
bottom § 579, p. 195 

Patterns for the moldings and roof pieces 
in the gables of an octagon pinnacle. 

§ 557, P- 170 
Patterns for the moldings and roof pieces 

in gablesof square pinnacle. .§ 554, p. 166 
Patterns for the volute of a capital .... 

§ 602, p. 220 
Patterns for elliptical curved moldings 

in a window cap, The § 569, p. 181 

Patterns for simple curved moldings in 

a window cap § 568, p. 179 

Patterns in a common window cap, The 

§ 437, P- 7o 

Patterns of a boss fitting over a miter 
in a molding, The § 596, p. 216 

Patterns of a coal hod g 500, p. 112 

Patterns of a cylinder (or pipe) and cone 
meeting at right angles to their axes. 

§ 527, P- 138 

Patterns of a cylinder joining a cone of 
greater diameter than itself at other 
than right angles § 531, p. 144 

Patterns of a cylinder mitering over the 
peak of a gable coping having a double 
wash, The § 505, p. 118 

Patterns of a finial the plan of which is 
octagon with alternate long and short 
sides, The § 591, P- 210 

Patterns of a flaring article, oblong in 
plan with rounded corners, and hav- 
ing greater flare at the ends than at 
the sides, The §477, P- 9° 

Patterns of a hip molding finishing a 
curved mansard roof which is square 
at the base and octagonal ut the top. . 

§ 584, P- 204 



Patterns of a hip molding upon a right 
atigle in a mansard roof mitering 
against the planceer of a deck cor- 
nice, The.. §575, P- 189 

Patterns of a hip molding upon an octa- 
gon angle of a mansard roof mitering 
against an inclined wash at the bot- 
torn, The § 577, p. Ig3 

Patterns of a hip molding upon an octa- 
gon angle in a mansard roof mitering 
against a bed molding of correspond- 
ing profile § 57tf, p. i g4 

Patterns of a frustum of a cone inter- 
secting a cylinder, their axes being at 
right angles, The § 528, p. 140 

Patterns of a molding mitering around 
an irregular four-sidid figure .§ 547, p. 159 

Patterns of an oblong tapering article 
with oue end square and one end 
semicircular, having more flare at 
the ends than at the sides, The. .tj 485, p. 98 

Patterns of an octagonal shaft, tho pro- 
file of which is curved, mitering upon 
the ridge of a roof, The § 597, p. 217 

Patterns of an octagonal vase. . .§ 439, p. 71 

Patterns of a square shaft to fit against 
a sphere, The §600, p. 220 

Patterns of a tapering article which is 
■square at the base and octagonal at 
the top, The §472, p. 87 

Patterns of a tapering article with equal 
fla re throughout, The § 494, p. 104 

Patterns of forms of the first class 

§405, p. 63 

Patterns of regular flaring ware, The 
usual method for describing. ..§454, p. 77 

Patterns of simple gable miters. § 548, p. ibo 

Patterns of two frustums of cones of un- 
equal diameter, intersecting at other 
than right angles to their axes, The. . 

§ 533, P- 148 

Patterns of the frustum of a cone join- 
ing a cylinder of greater diameter 
than itself at other than right angles, 
the axis of the frustum passing to one 
side of axisof cylinder, The. .§ 530, p. 143 

Patterns of the moldings upon a panel, 
the shape of which is a scalene tri- 
angle, The §545, p. 1 58 

Patterns of two cones of unequal diam- 
eter intersecting at right angles to 
their axes, The § 532, p. 145 

Peak of a gable coping having a double 
wash, The patterns of a cylinder mi- 
tering for the § 505, p. 118 

Pedestal square in plan, The pattern 
for a § 586, p. 207 

Pedestal, The pattern for an octagonal. 

§ 590, p. 209 

Pedestal, The pattern for a, tho plan of 
which is a hexagon § 588, p. 208 

Pedestal, The patterns fora, of w Inch the 
pi in is an equilateral triangle. § 5S5, p. 207 

Pediment, A .§ 177, p. 12 

Pediment, A raking bracket in a curved 

§ 573, P- 187 

Pediment, In a broken, to ascertain tho 
profile of the horizontal return at tho 
top, together with its miters. .§ 551, p. 163 

Pencil, Drawing an ellipse with string 
and § 377, P. 5 6 

Pencil, Drawing an ellipse to specified 
dimensions with a string and. .§ 378, p. 56 

Pencils, Lead § 253-257, P- 24 

Penetration of solids § 113, p. 9 

Pens, Drawing §258-261,^1. 24-25 

Pentagon § 47, p. 4 

Pentagon, To draw a regular, upon a 
given side § 325, P- 42 

Pentagon, To draw a regular, withm a 
given circle § 314. P- 39 

Pentagon, The patterns for a vase tho 
plan of which is a § 587, p. 207 



238 



Index. 



Pentagonal pyramid § 102, p. 8 

Pentagonal prism, A § 93, P- 7 

Perimeter . . ..§ 54, p. 4 

Perpendicular at a given point in a 
straight line, To erect a § 279, p. 30 

Perpendicular at or near the end of a 

given straight line, To erect a 

§ 280-281, p. 30 

Perpendicular lines § 16, p. 2 

Perpendicular to another line, To draw a 

§ 282, p. 30 

Perpendicular, To raise a, to an arc of a 
circle without having recourse to the 
center § 294, p. 34 

Perspective, A § 182, p. 13 

Pieces, The pattern for a vase or urn in 
any number of § 440, p. 72 

Pilaster, A § 176, p. 12 

Pinnacle, A § 175. P- I2 

Pinnacle, Patterns for the moldings and 
roof pieces in the gables of an octa- 
gon § 557, P- 170 

Pinnacles, Patterns for the moldings 
and roof pieces under the gables of 
square §^554- P- ID0 

Pins, Drawing § 268, p. 26 

Pipe and cone meeting at right angles to 
their axes, The patterns of a cylinder 
or § 527, P- 138 

Pipe, A thieepiece elbow in tapering. . 

§517, p. 129 

Pipe, A three-piece elbow in flaring, the 
middle section of which is straight. . . 

§518. p. 130 

Pipe, A two-piece elbow in elliptical. . . 

§ 520, p. 133 

Pipe, A two-piece elbow in tapering. . . . 

§ 516, p. 127 

Pipe carried around a semicircle by 
means of cross joints, A § 514. p. 125 

Pipe to fit over the ridge of a roof, The 
pattern of a round § 504, p. 117 

Pipe, The pattern of an elliptical, to fit 
against roof of one inclination. § 501, p 116 

Pipe, The pattern of a round, to fit 
against roof of one inclination. . § 501, p. 1 16 

Pipes of different diameters, A joint at 
other than right angles between two, 
the axis of the smaller pipe being 
placed to one side of that of the larger 
one § 5=6, P- 138 

Pipes of different diameters, A joint be- 
tween two, intersecting at other than 
right angles § 525, p. 137 

Pipes of different diameters, A T-joint 
between § 522, p. 134 

Pipes of different diameters, A T joint 
between, the smaller pipe setting to 
one side of the larger § 523, p. 135 

Pipes, of the same diameter, A joint be- 
tween two, at other than right angles 

§ 5=4, P. 136 

Pipes of the same diameters, AT- joint 
between § 521, p. 133 

Plain dividers § 238, p. 20 

Plain frieze _.§ 128, p. 10 

Plain surface, A butt miter against a. .. 

§ 539, P- 154 
Plain scales, Method of constructing. . . 

§ 248, p. 22 
Plain surface oblique in plan, A butt 
miter of a molding inclined in eleva- 
tion against a § 54 r > P- J 55 

Plain surfaces of mansard finish are de- 
veloped, The principles upon which 

the § 574, P- 189 

Plain tapering keystone, The patterns 

for the face and side of a. . . .§ 570, p. 182 
Plain window cap and its several pat- 
terns, A § 567, p. 17S 

Plan, A § 180, p. 12 

Planceer § 131, p. 10 

Planceer of a deck cornice, The pattern 



of a hip molding upon a right angle in 
a mansard roof mitering against the. 

§ 575, P- 189 
Planceer of a deck cornice upon a man- 
sard roof which at the base is square 
and at the top octagon, Patterns of a 
hip molding mitering against.. § 580, p, 19S 

Plane, A § 22, p. 2 

Plane figure, A § 23, p. 2 

Plastic material by means of a stay, 

Developing a molding in § 410, p. 64 

Plunge bath §4S5, fig- 329 

Point, A § 4, p. 1 

Point, A given .§ 9, p. 1 

Points, Use of the T-Square in dropping, 

§439, P- 7i 

Polygon § 25, p. 2 

Pohgon, Adjusting the drawing of a. to 
suit the requirements of miter cutting 

§ 3 6 9-373, P- 55 
Polygon, General rule for drawing a 
regular, the length of a side being 

given § 334, P- 44 

Polygon in a circle, General rule for 

drawing any regular § 322, p. 41 

Polygon of eleven sides within a given 

circle, To draw a § 323, P- 4 1 

Polygon, Regular § 25, p. 2 

Polygon, To construct a regular, of thir- 
teen sides, the length of a side being 

given § 335, p. 44 

Polygons, The construction of regular, .p. 39 
Polygons, Tables of divisions upon the 
square to be used in constructing. . . . 

§ 368, p. 54 
Polygons, Construction of regular, by 

the use of a carpenter's square p. 53 

Polygons, Construction of regular by 

means of the protractor p. 5 1 

Polygons, Construction of regular, by 
the use of the T-square, triangles or 

set squares p. 45 

Premise § 195, P- '3 

Preparing india ink for use § 266, p. 26 

Principles, Skill dependent upon a know- 
ledge of § 276, p. 29 

Principles upon which the plain surfaces 
of a mansard finish are developed, 

The §574, P. 189 

Prism, A § 90-94 p. 7 

Problem 1 191, p. 13 

Problems, Arrangement of § 276, p. 29 

Problems, Geometrical p. 29 

Problems, Pattern p. 81 

Profile, A raked § 161, p. 12 

Profile, A molding described as a form 

or surface generated by a § 408, p. 64 

Profile of a corresponding horizontal 
molding, From a given profile in an 
inclined molding to establish the, to 
miter with it at an octagon angle in 
plan, and the miter patterns involved. 

§ 553, P- 166 
Profile of horizontal return at the top, 
together with its miters, In a broken 
pediment to ascertain the. . .§ 551, p. 163 
Profile, To ascertain the, of a horizontal 
molding adapted to a miter, with a 
given inclined molding at right angles 
in plan, and the several miter patterns 

involved § 549, p. 160 

Profile, to establish the, for a corre- 
sponding inclined molding, From a 
given horizontal profile, tomiter with 
it at an octagon angle in plan, and sev- 
eral miter patterns involved. .§ 552, p. 165 

Profile, The normal § 162, p. 12 

Profiles, Miter between two moldings of 

different § 544, p. 156 

Project a figure, To § 200, p 14 

Projection of a circle, An oblique § 3S8, p. 57 
Projection of a circle, To draw an ellipse 
as the oblique § 389, p. 58 



82 



82 



Prolong § 205, p 14 

Properties of an elliptical cone..§ 452, p. 76 

Proposition § 1 92, p. 13 

Protractor, The. . . .§ 244-246, 356, p. 22, 51 
Protractor, Construction of regular poly- 
gons by means of the p. 51 

Protractor, The divisions of the.. § 357, p. 51 

Protractor, Using the § 3^8, p. 51 

Pudding Molds §488, fig. 333 

Pyramid. . § 102, p. 8 

Pyramid revolved in such a manner as 
to describe the shape of its covering, 

A hexagonal § 446, p. 74 

Pyramid, Theenvelopeof asquars.§462, p. 81 
Pyramid, The envelope of a triangu- 
lar § 461, p. 8r 

Pyramid, The envelope cf a hexago- 
nal §464, P. 

Pyramid, The envelope of the frustum 

of a square § 463, p 

Pyramid, The envelope of the frustum 
of an octagonal, having alternate long 

and short sides § 466, p. 83 

Pyramid, The envelope of the frustum 

of an octagonal § 465, p. 82 

Pyramid which is diamond shape in 
plan, The envelope of the frustum 

of a §467, p. 84 

Quadrangular prism, A § 92, p. 7 

Quadrangular pyramid § 102, p. 8 

Quadrant, A § 65, p. 5 

Quadrilateral figure, A § 39, P- 3 

Quality of India ink § 264, p. 25 

Quarter circumference of a given circle, 
To draw a straight line equal to 

the § 299, p. 3i 

Radii § 59, p. 4 

Radius § 59, P- 4 

Raise, To § 203. p. 14 

Raised cover, Pattern of a, fitting oblong 

vessel with round ends § 476, p. 90 

Rake a molding, To § 159, p. 12 

Rake moldings § 159, p 

Raked miter, A § 166, p 

Raked molding § 160, p 

Raked profile, A § 161, p 

Raked stay, A ■ § 161, p 

Raking bracket in s curved pediment, 

A §573, P 

Raking bracket, Patterns for a 

§ 572, p. 184 

Range tea kettles § 488, fig. 333 

Ratio § 201, p. 14 

Rectalinear figures § 24, p. 2 

Rectangle : §45, P- 4 

Rectangle, A right cylinder generated 

by the revolution of the § 441, p. 72 

Rectangle, To draw an approximate 
ellipse in a given, by means of inter- 
secting lines § 394, p 

Rectangle, To draw an ellipse within a 
given, by means of intersecting lines. 

§ 393, P 

Rectangular article, The pattern of a, 

three sides of which are vertical, the 

fourth being inclined § 469, p. 

Rectangular flaring article having one 
end upright, The pattern of a.§ 470, p, 
Rectangular flaring article, The pattern 

of a §468, p. 84 

Regular decagon upon a given side, To 

draw a § 331, p. 43 

Regular decagon within a given circle, 

To draw a § 319, p. 40 

Regular dodecagon upon a given side, To 

draw a § 333, p. 44 

Regular dodecagon within a given cir- 
cle, To draw a § 321, p. 41 

Regular flaring article, The pattern of a, 
which in shape is oblong with semi- 
circular ends § 475, p. 89 

Regular flaring elliptical frustum is a 
part, The solid of which a § 458, p. 80 



12 
12 

12 
12 

187 



60 
59 

85 
85 



Regular flaring oblong article, The pat- 
tern of a, with round corners.. § 474, p 

Regular flaring ware corresponds, Anal- 
yses of the solid to the envelope of 
which §455,p 

Regular flaring ware, Composition of 
tbe shape of g 458, p 

Regular flaring ware corresponds, The 
string or thread method for develop- 
ing the envelope of the solid to which. 

_ , _ . §457.P 

Regular flaring ware corresponds, The 
solid to the envelope of which. § 453, p 

Regular flaring ware, The usual method 
for describing patterns of § 454, p 

Regular heptagon upon a given side. To 
draw a § 328, p. 

Regular heptagon within a given circle, 
To draw a § 316, p 

Regular hexagon upon a given side, To 
draw a g 327, p 

Regular hexagon within a given circle, 
To draw a § 315, p 

Regular nonagon upon a given side, To 
draw a § 330, p 

Regular nonagon within a given circle, 
To draw a § 3 [3, p 

Regular octagon upon a given side, To 
draw a § 329, p 

Regular octagon within a given circle, 
To draw a § 317, p 

Regular octagon within a given square, 
To draw a § 336, p. 

Regular polygon § 25, p 

Regular polygon, General rulefor draw- 
ing, length of side beinggiven.§ 334, p. 

Regular polygon in a circle, General 
rule for drawing § 322, p. 

Regular polygon of eleven sides within 
a given circle, To draw a. . . .§ 323, p. 

Regular polygon of thirteen sides the 
length of a side being given. ..§ 335, p. 

Regular polygons, Construction of, by 
the use of the T-square, triangles or 
set square p 

Regular polygons, Construction of, by 
means of the protractor p 51 

Regular polygons, The construction of .p 

Regular polygons, Construction of, by 
the use of a carpenter's square p 

Regular pentagon upon a given side, To 
draw a §3-5, P 

Regular pentagon within a given circle 
To draw a § 314, p 

Regular undecagon upon a given side, 
To draw a § 332, p 

Regular undecagon within a given cir- 
cle, To draw a § 320, p 

Requirements for pattern cutting §425, p 

Requirements of miter cutting, Adjust- 
ing the drawing of a polygon to suit 
the § 369-373 P- 55 

Return miter, A square, or a miter at 
right angles, as in a cornice at the cor- 
ner of a building § 538, p. 

Return at the top, together with its 
miters, In a broken pediment to ascer- 
tain profile of horizontal § 551, p. 

Return miter at other than a right 
angle, as in a cornice at the corner of 
a building, A § 539, P- T 53 

Return miter, The usual method of de- 
veloping a square § 429, p 

Returned miter, A square § 428, p 

Reverse stay, A § 408, p 

Revolution, A solid of § 441, p 

Revolution, Covering of a triangular 
frustum obtained by a § 448, p 

Revolution of a right-angled triangle, A 
right cone generated by the. ..§ 441, p. 72 

Revolution of a right cone by which the 
shape of its envelope is described, 
The §443, P- 73 



79 



77 



77 



42 



40 



42 



39 



43 



40 



43 



4'J 



41 



41. 



II 



45 



39 



53 



4- 



153 



163 



Index. 

Revolution of solids, The § 443, p. 73 

Revolution of the rectangle, A right 

cylinder generated by the. . . .§ 441, p. 72 
Revolution, The covering of a cube de- 
veloped by g 448, p. 75 

Rhomb §43, p. 3 

Rhomboid g 44, p. 3 

Rhombus § 43, p. 3 

Ridge capping g i 5 g j p. IT 

Ridge Molding, A g 155, p. n 

Ridge of a roof. The pattern of a flange 

to fit around a pipe and over g 50S, p. 120 
Ridge of a roof, The pattern of an octa- 
gon shaft fitting over the. . . .§ 503, p. 1:7 
Ridge of a roof, The patterns of an octa- 
gonal shaft, the profile of which is 
curved, mitering upon the. . .§ 597. p. 217 
Ridge of a roof, The pattern of a round 

pipe to fit over the § 504, p. 117 

Ridging § 155-156, p. n 

Right angle, A § 18, p. 2 

Right-angled triangle, A right cone 
generated by the revolution of a 

§ 441, p. 72 

Right-angled triangle § 30, p. 3 

Right cone, A g 98, p. 8 

Right cone contained between planes 
oblique to its axis, The envelope of a 
frustum of a scalene cone or the en- 
velope of the section of a. . . .§ 49T, p. 102 
Right cone generated by the revolution 
of a right-angled triangle, A. .§ 441, p. 72 

Right cone, Frustum of a g 450, p. 75 

Right cone, The envelope of a. . -§486, p. 99 
Right cone, The envelope of a frustum 

of a g 4S8, p. 100 

Right cone, The envelope of a, from 
which a section is cut parallel to its 

axis § 487, p. 99 

Right cone, The envelope of the frustum 
of a, the upper plane of which is ob- 
lique to its axis g 489, p. 100 

Right cone, The revolution of a, by 
which the shape of its envelope is de- 
scribed g 443, p. 73 

Right cylinder generated by the revolu- 
tion of the rectangle, A § 441, p. 72 

Right pyramid, A § 103, p. 8 

Roof of one inclination, A conical flange 
to fit around a pipe and against a. . . . 

§507, P- "9 

Roof of one inclination. The pattern of 
an elliptical pipe to fit against the . . . 

§ 502, p. 116 

Roof of one inclination, The pattern of 
flange to fit around the pipe and 
against the § 506, p. 119 

Roof of one inclination, The pattern of 
a round pipe to fit against the. g 501, p. 116 

Roof pieces, Patterns for the molding 
and, in the gables of an octagon pin- 
nacle g 557, p. 170 

Roof pieces under the gables of the 
square pinnacles. Patterns for the 
moldingand § 554, p. 166 

Roof, The pattern of a flange to fit 
around a pipe and over the ridge of a 

§'508, p. 120 

Roof, The pattern of an octagon shaft 
fitting over the ridge of a. . . .§ 503, p. 117 

Roof, The pattern of a round pipe to fit 
over the ridge of a § 504, p. 1 1 7 

Round at the top, elliptical at the base, 
The pattern of an irregular flaring 
article which is, the top being so situ- 
ated with respect to the base as to be 
tangent to one end of it when viewed 
in plan § 496, p. 107 

Round at the top, The pattern of a 
flaring article square at the base and 

§ 473, P- 87 

Round corners, The pattern of a regular 
flaring article with § 474, P- 89 



90 

28 
27 



239 

Round ends, Pattern of a raised cover 
fitting an oblong vessel with, .g 476, p. 90 

Round flaring pans § 4S8, fig. 333 

Round pipe to fit over the ridge of a 
roof, The pattern of a g 504, p. 117 

Round pipe, The pattern of a, to fit 
against a roof of one inclination... 

„ , § 501, P- 116 

Round top, The pattern of an article 
having an elliptical base and a.§483, p. g6 

Round top, The pattern of an oblong 
flaring article having a g 4S2, p. 95 

Rounded corners and having greater 
flare at the ends than at the sides, 
The pattern of a flaring article oblong 
in plan with g ^y : p 

R >yal (drawing paper; g 274, p. 

Rubber, India g 270-271, p. 

Rubbing down India ink g 266, p. 26 

Rule for avoiding errors in miter cut- 
ting §43-, p. 70 

Rule for cutting a square miter, The 
short ." g 427, p. 68 

Rule for cutting square miters, Identity 
of long and short g 438, p. 70 

Rule for drawing any regular poljgon, 
the length of a side being given, Gen- 

p^t.- ■■;•••: • • • •§ 334, P- 44 

Kule tor drawing any regular polygon 

in a circle, General g 322, p 

Rule for pattern cutting, General. . . . 

T> , • • S 426 P 

Kules in pattern cutting g 459, p 

1 Scale by which to divide a straight line 
into any number of equal parts 

„ , ' , § 28 7, P- 32 

scale drawing, A g 187, p. 13 

Scale scoop, The patterns for a. g 497, p 107 

Scales g 247-252, pp. 22-23 



41 



68 
80 



.8 



8 99, P- 



Scalene cone, A. . 

Scalene cone, The envelope of a.g 490, p. ioi 

Scalene cone, The envelope of a frustum 

of a, or the envelope of the section of 

a right cone contained between planes 

oblique to its axis g 491, p. 102 

Scalene triangle g 29, p. 3 

Scalene triangle, The patterns of the 
moldings upon a panel, the shape of 

which is a g 545, p. 158 

Science of pattern cutting, The art 

and p 63 

Scoop, The pattern for a scale. g 497, p. 107 
Scroll to a specified width, as for a 

bracket or modillion, To draw .g 310, p. 38 
Seams, To form a semicircle in a pipe by 

means of longitudinal § 515, p. 126 

Secant, A g 77, p. 6 

Seconds § 12, p. 5 

Section, A , §181, p. 12 

Section of a cone through its opposite 
sides, To describe the shape of an ob- 
lique § 391, p. 59 

Section of a cone, To draw an ellipse as 

a § 39 1 - P- 59 

Section of a cylinder, To describo the 

form or shape of an oblique. . .§ 389, p. 58 
Section of a right cone contained be- 
tween planes oblique to its axis. The 
envelope of a frustum of a scalene 
cone or the envelope of the. . .§491, p. 102 
Section through an elliptical cnne, An 

irregular § 49S, p. 108 

Sector g 64, p. 5 

Segment g 62, p. 4 

Segment of a circle by a triangular 
guide, the chord and hight being 

given, To strike a § 292, p. 34 

Segment of a circle, The chord and hight 
of a, being given, to find the center 

by which the arc may be struck 

§ 290, p. 33 

Segmental pediment, A fig. 92, p. 12 

Semicircle S 6 1 , p. 4 



240 



Index. 



Semicircle, A pipe carried around a, by 
m?ans of cross joints £ 514, p. 125 

Semicircle in a pipe by means of longitu- 
dinal seams, To form a § 515, p, 126 

Semicircular ends, The pattern of a flar- 
ing article, tbe top of which is round 
and the bottom of which is oblong, 

with § 4So, p. 93 

Semicircular ends, The pattern of a reg- 
ular flaring article which in shape is 
oblong with § 475, p. 89 

Semi circumference of a circle, To draw 
a straight line equal to the. . . .§ 298, p. 35 

Set square, The governing principle of 
tbe § 339, p. 45 

Set square, The use of a, in dividing a 
circle § 339, p. 45 

Set square?, Calculation of angles by 
means of § 339, P 45 

Set squares for dividing circles 

§ 337-342, pp. 45-46 

Set squares, Home-made § 235, p. 20 

Set squares, Names of § 340, p. 4b 

Set squares, Testing § 34, p. 20 

Set squares, Triangles or § 233, p. 19 

Shaft to fit against a ball, To describe 
the patterns of an octagon . . .§ 601, p. 220 

Shaft to fit against a sphere, The pat- 
terns of a square § 600, p. 220 

Shaft, The pattern of an octagon, fitting 
over the ridge of a roof § 503, p. 117 

Shape and extent of its covering, A 
cylinder revolved showing the. ^447, p. 74 

Shape in thepatternof an opening nia a e 
in its side, Cylinder revolved showing 
the" §447, P- 74 

Shape of its covering, A hexagonal 
pyramid revolved in such a manner 
as to describe the § 446, p. 74 

Shape of its envelepe, A scalene cone 
revolved to show the § 445, p. 73 

Shape of its envelope, A cylinder with 
one end cut obliquely and revolved so 
as to show the § 447, p. 74 

Shape of its envelope is described, The 
revolution of a right cone by which 
tbe §443, F- 73 

Shape of regular flaring ware, Composi- 
tion of the § 45S, p. 80 

Shapes entering into tinware. . . §406, p. 64 

Shapes in pattern cutting, Conical 

§ 449, P- 75 
Sharpening of lead pencils, The.§ 256, p. 24 

Sheet metal pattern cutting § 3, p. 1 

Ship ventilator, The patterns ior a, 

having an oval mouth on a round 

pipe § 605, p. 223 

Short and long methods of cutting a 

square miter compared§ 451-433, pp. 6S-69 
Short rule for cutting a square miter, 

The §427, p. 68 

Short rule for cutting square miters, 

Identity of long and § 438, p. 70 

Side being given, To construct an 

equilateral triangle, the length of a . . 

§ 365, P- 53 
Side being given, To construct a hexagon 

by means of six equilateral triangles, 

the length of a § 367, p. 54 

Side, To construct a hexagon the length 

of a, being given § 366, p. 53 

Side, To draw an equilateral triangle 

upon a given § 347, p. 4S 

Side, To draw an octagon upon a given. 

§ 350, p. 48 
Side, To draw a square upon a given. . . 

§ 348, 355, PP- 43, 51 
Side, To draw a hexagon upon a given . 

§ 349, P- 4S 
Side, The length of a, being given, to 
construct a regular polygon of 13 sides 

by the general rule § 335, p. 44 

Side, The length of a, being given, gen- 



eral rule by which to draw any regu- 
lar polygon. §334, P- 44 

Side, Upon a given side to construct an 

equilateral triangle § 324, p. 41 

Side, Upon a given, to draw a regular 

decagon § 331, p. 43 

Side, Upon a given, to draw a regular 

dodecagon § 333, p. 44 

Side, Upon a given, to draw a regular 

heptagon § 328, p. 42 

Side, Upon a given, to draw a regular 

hexagon § 327, p. 42 

Side, Upon a given, to draw a regular 

octagon § 329, p. 43 

Side, Upon a given, to diaw arfgular 

nonagon § 330, p. 43 

Side, Upon a given, to draw a regular 

pentagon § 326, p. 42 

Side, Upon a given, to draw a regular 

uudecagon § 332, p. 43 

Sides of a triangle , . .§ 36, p. 3 

Sides, The length of three, being given, 

to construct a triangle § 325, p. 41 

Similarity between the measurements 
necessary to the development of a couo 
pattern and a miter pattern. .§ 444, p. 73 

Sine of an arc, The § 1 9, p. 6 

Sink, A § 138, p. 11 

Sink in face, Patterns for a keystone 

with § 571, p. 182 

Sizes of drawing paper § 274, p. 28 

Skill dependent upon knowledge of 

principles § 276, p. 29 

Slabs for India ink g C67, p. 26 

Slop jars § 4S8, fig. 333 

Soapmaker's float, Patterns for.g 537, p. 151 

Soffit ..§ 112, p. ro 

Solid, A §89, p. 7 

Solid, Analyses of the, to the envelope 
of which regular flaring ware corres- 
ponds § 455, p. 78 

Solid of revolution, A § 441, p. 72 

Solid of which a regular flaring elliptical 

frustum is a part, The § 458, p. 80 

Solid, The striDg or thread method for 
developing the envelope of the, to 
which regular flaring ware corres- 
ponds 



.S 



§457, P- 79 
Solid to the envelope of which regular 

flaring ware corresponds, The § 453, p. 77 

Solids, Intersection of § 114, p. 9 

Solids, Penetration of § 113, p. g 

Solids, The revolution of § 443, p. 72 

Spacers § 239. p. 20 

Sphere, A § 101, p. 8 

Sphere, The patterns of a square shaft 

to fit against a § £00, p. 220 

Spiral by means of a spool of thread, To 

draw a § 309, p. 38 

Spiral from centers with compasses, To 

draw a § 30S, p. 3S 

Spire mitering upon eight gables, The 

pattern of a conical § 563, p. 174 

Spire mitering upon eight gables, The 

pattern of an octagon § 562, p. 174 

Spire, The pattern of a conical, mitering 

upon four gables § 560, p, 172 

Spire, The pattern of an octagon, miter- 

upon four gables § 561, p. 173 

Spire, The pattern of a square, mitering 

upon four gables § 559, p. 171 

Sponge bath § 4SS, fig. 333 

Spouts, Tea kettle § 529, fig. 395 

Square § 46, p 4 

Square about a circle, Todrawa.§ 354, p. 50 
Square and a strip of wood, To draw an 

ellipse of given dimensions by means 

of a § 386, p. 57 

Square at the base and octagonal at the 

top, The patterns of a tapering article 

which is § 472, p. 87 

Square at the base and round at top, The 

pattern of a flaring article. . .§ 473, p. 87 



68 



68 



Square, Construction of regular polygons 
by the use of a carpenter's p. 53 

Square face or panel miter, A. ..§ 434, p. 69 

Square in plan, The pattern for a ped- 
estal § 586, p. 207 

Square mansard roof, Patterns for a hip 
molding upon a, mitering against bed 
molding at the top § 576. p. igi 

Square miter § 167. p. 12 

Square miter, A miter line not usually 
employed in cutting a § 427, p 

Square miter, Abbreviated method for 
cutting a § 427, p 

Square miter. Short and long methods of 
cutting compared. . . .§ 431-433, pp. 68-69 

Square miter, Method of cutting a 

§ 427-435, IP- t8-70 

Square miter, The short rule for cutting 

a §427, P- 68 

Square miters, Identity of long and short 

rule for cutting § 438, p. 70 

Square pans § 46S, fig. 295 

Square pinnacles, Patterns for the mold- 
ing* and roof pieces forming Ihe gables 

of § 554, p. 166 

Square pyramid, Envelope of a. §462, p. 81 
Square pyramid, Envelope of the frus- 
tum of a § 463, p. 82 

Square return miter, or a miter at right 
angles, as in a cornice at the corner of 

a building, A § 538, p. 153 

Square return miter, The usual method 

of developing a § 429, p 86 

Square return miter, A § 428, p. 68 

Square shalt to lit against a sphere, The 

patterrs of a § 600, p. 220 

Square spire, The pattern of a, mitering 

upon four gables § 559, p. 171 

Square, To draw a, within a given ciicie 

§ 313, P 
Square, To find the center from \\ hich 
a given arc is struck by the use of 

the § 291, p. 

Square, The gore piece forming transi- 
tion between an octagon and a. § 565, p. 176 
Square, The gore piece in a molding 
forming transition from an octagon 

toa § 504, P- 175 

Square, Table of divisions upon the, to 

be used in constructing polygons. § 368, p 54 

Square, The steel § 229, 232, p 19 

Square upon a given side, To draw a . . . 

§348,355, PP- 48, 51 
Square within a given circle, To draw a 

§ 344, 361, pp. 47, 52 
Square, To draw a regular octagon with- 
in a given § 336, p. 43 

Stay, A § 408, p. 64 

Stay, A raked § 161, p. 12 

Stay, Developing a molding in plastic 

material by means of a § 410, p. 64 

Stay, A molding is made by bending 

metal to fit a given § 407, p. 64 

Stay, outside, An §409. p. 64 

Stay of amoldiug § 158, p. 12 

Stay, reverse, A § 408, p. 64 

Stay, The normal g 162, p. 12 

Steel edges for drawing tables. .§ 211, p. 16 

Steel square, The § 229-232, p. 19 

Steel squares, Testing § 230-232, p. 19 

Stew pans, Oyster § 4S8, fig. 333 

Stop block, A § 140, p. 1 1 

Straight edges § 221-224, p. 18 

Straight line, A § 6, p. 1 

Straight line equal to any given part of 
of a circle less than a semicircle, To 

draw a § 300-301, p. 35 

Straight line equal to the circumference 

of a circle, To draw a.§ 296-297, pp. 34-35 
Straight line equal to the semi-circum- 
ference of a circle, To draw a.§ 298, p. 35 
Straight line equal to the quarter circum- 
ference of a circle, To draw a.§ 299, p. 35 



39 



33 



Index. 



241 



Straight line into two equal parts, To 

divide a given . .§ 283-2S5, p. 31 

Straight line into any number of equal 

parts, To divide a. . ..§ 286-287, PP- 3* _ 32 
Straight line parallel to a given line, To 

draw a § 277, p. 29 

Strainers, Milk § 488, fig. 333 

Stretchout lines 8 413, p. 66 

Stretchout line, Position of the. .§ 419, p. 67 
String and pencil, Drawing an ellipse to 

specified dimensions with a. . .§ 378, p. 56 
String or thread method for developing 
the envelope of the solid to which 
regular flaring ware corresponds, 

The : §457, P 79 

String and pencil, Drawing an ellipse 

with § 377, p. 56 

Students of pattern cutting, General 

advice to § 459, p. 80 

Succession of parallel forms, A molding 

is a §407, P- 64 

Super royal (drawing paper). . . .§ 274, p. 28 
Supplement of an arc or angle. . . .§ 75, p. 6 

Surface, A § 2r, p. 2 

Surface, A molding terminating against 

an irregular § 421. p. 67 

Surface generated by a profile, A mold- 
ing described as a form or. . . .§ 408 p. 64 
Surfaces, Miter cutting as a system of 

measurements upon £ 436, p. 70 

Surfaces of mansard finish are devel- 
oped, The principles upon which the 

plain § 574, P- 189 

System of measurements upon surfaces, 

Miter cutting is a § 436, p. 70 

Table of divisions upon the square to be 
used in constructing polygons. § 368. p. 54 

Tables, Drafting g_ 208, p. 15 

Tables, Testing drawing § 217, p. 17 

Tables, Woods for drawing § 210, p. 15 

Tacks, Thumb 8 268, p. 26 

Talk about the ellipse, Familiar. § 374, p. 55 

Tangent, A §67, 76, pp. 5-6 

Tangent of an angle or of an arc.§ 76, p. 6 
Tangent, To draw a, to a circle or pat- 
tern of a circle without having re- 
course to the center § 295, p. 34 

Tapering article, The patterns of a, 
which is square at the base and oc- 
tagonal at the top §47 2 , P- 8 7 

Tapering article with equal flare through- 
out, The patterns of a § 494, p. 104 

Tapering article with one end square 

and one end semicircular, having more 

flare at the ends than at the sides, 

The patterns of an oblong. . . .§ 485, p. 98 

Tapering horn, The patterns of a curved, 

octagonal in section § 605, p. 225 

Tapering keystone, The patterns for the 

face and side of a plain § 570, p. 182 

Tapering pipe, A two-piece elbow in . . . 

§ 516, p. 127 
Tapering pipe, A three-piece elbow in. . 

§ 517, P- 129 

Tea kettle spouts § 529, fig. 395 

Tea kettles, Flaring § 433, fig. 333 

Tea kettles, Range § 488, fig. 333 

Tea pot, Flaring § 4S8, fig. 333 

Tea pot, Octagon § 4 66 , fi g s - 289-291 

Technicalities, Definitions and p. I 

Testing drawing boards and tables 

§ 217-220, p. 17 

Testing set-squares § 234, p. 20 

Testing steel squares § 230-232, p. 19 

Tetrahedron, A. § 106, p. 8 

Thread method for developing the en- 
velope of the solid to which regular 
flaring ware corresponds, The string 

or :§457, P- 79 

Three-piece elbow, A § 5 T °> P- I2 I 

Three-piece elbow in flaring pipe, the 
middle section of which is straight, 
A §5l8, p. 13" 



Three-piece elbow in tapering pipe, A. . 

§ 517, P- 129 
Three piece elbow, the middle section of 

which tapers § 519, p. 131 

Three sides of which are vortical, the 
fourth being inclined, The pattern of 

of a rectangular article § 469, p. 85 

Thumb tacks § 268, p. 26 

Tinware, Shapes entering into. .9 406, p. 64 
T-joint between pipes of different, diam- 
eters, A § 522, p. 134 

T-joint between pipes of different diam- 
eters, the smaller pipe setting to one 

side of the larger, A § 523, p. 135 

T-joint between pipes of the same diam- 
eters, A § 521, p. 133 

To ascertain the profile of a horizontal 
molding adapted to miter with a 
given inclined molding at right angles 
in plan, and the several miter patterns 

involved §' 549, p. 160 

To construct a ball in any number of 
pieces in the general shape of gores. . 

§ 599. P- 218 

To construct a ball in any number of 
pieces the general shape of zones. . . . 

§ 598. P- 218 

To construct an ellipse to given dimen- 
sions by the use of two circles and in- 
tersecting lines § 392, p. 59 

To construct an equilateral triangle, the 
length of a side being given. .§ 365. p. 53 

To construct an equilateral triangle upon 
a given side § 324, p 41 

To construct a hexagon by means of six 
equilateral triangles, the length of a 
side being given § 367, p. 54 

To construct a hexagon, the length of a 
side being given § 366, p. 53 

To construct a regular polygon of thir- 
teen sides, the length of a side being 
given § 335, P- 44 

To construct a triangle, the lengths of 
three sides being given § 325, p. 41 

To describe an Ionic volute g' 307, p. 37 

To describe the form or shape of an 
oblique section of a cylinder. .§ 3S9. p. 58 

To describe the pattern of an octagon 
shaft to fit against a ball. . . .§ 601, p. 220 

To describe the shape of an oblique sec- 
tion of a cone through its opposite 
sides 3 391 ' p ' 5q 

To develop a pattern g 19S, p. 13 

To divide an arc of a circle into any 
given number of equal parts. .§ 302, p. 36 

To divide a given angle into two equal 
parts .§ 2S8, p. 32 

To divide a given straight line into two 
equal parts with the compasses by 
means of arcs § 2S3, p. 31 

To divide a giveu straight line into any 
number of equal parts § 2S6, p. 31 

To divide a straight line into two equal 
parts by the use of the triangle or set 
square § 2S5. p. 31 

To divide a straight line into two equal 
parts by the dividers § 284, p. 31 

To draw an approximate ellipse in a 
given rectangle by means of inter- 
secting lines •§ 394, P- 6o 

To draw an approximate ellipse with 
the compasses to given dimensions, 
using two sets centers § 397-398, P- 6° 

To draw an approximate ellipse with the 
compasses to given dimensions, using 
three sets of centers § 399, P- 6l 

To draw a circle through three given 
points not in a straight line. . .§ 293, p. 34 

To draw a dodecagon within a given 
circle § 3&3i P- 52 

To draw an egg-shaped figure. .§ 402, p. 62 

To draw an ellipse as a section of a 
cone... § 39 1 . P- 59 



To draw an ellipse as the oblique pro- 
jection of a circle § 389, p. 58 

To draw an ellipse of given dimensions 
by means of a square and a strip of 
wood § 386, p. 57 

To draw an ellipse to given dimensions 
by means of a trammel § 3S4, p. 57 

To draw au ellipse to specified dimen- 
sions with a string and pencil. § 378, p. 56 

To draw an ellipse within a given 
rectangle by means of intersecting 

lines 8 393, P- 59 

To draw an elliptical figure with the 

ci mipasses, the length only being 

given § 395, p. 60 

To draw an equilateral triangle about a 

given circle §351, p. 49 

To draw an equilateral triangle upon a 

given side § 347, p. 48 

To draw an equilateral triangle within 

a given circle.g" 312, 343, 360, pp. 39, 46, 51 

To draw a Grecian ogee g* 304, p. 36 

To draw a hexagon about a given circle 

m , , § 35=, !'• 49 

To draw a hexagon upon a given side . . 

§ 3-19, P- 48 
To draw a hexagon within a given circle, 

§ 345, P. 47 
To draw a line parallel to another by the 

use of triangles or set squares § 27S, p. 30 
To draw a Hue perpendicular to another 
line by the use of triangles or set 

squares § 2S2, p. 30 

To draw an octagon about a given circle 

§353, P- 5° 
To draw an octagon upon a given side. 

§ 350, P- 43 
To draw an octagon within a given 

circle § 346, 362, pp. 47, 52 

To draw an ogee by means of two quar- 
ter circles § 303, p. 36 

To draw a parabola by the intersection 

of lines, its hight and baseorordinato 

being given § 305, p. 36 

To draw a regular decagon upon a given 

side § 33r, p. 43 

To draw a regular decagon within a 

given circle § 319. p. 40 

To draw a regular dodecagon upon a 

given side § 333, p. 44 

To draw a regular dodecagon within a 

given circle § 321, p. 41 

To draw a regular heptagon upon a 

givnn side § 328, p. 42 

To draw a regular heptagon within a 

given circle § 316, p. 40 

To draw a regular hexagon within a 

given circle § 315, p. 39 

To draw a regular hexagon upon a 

given sido § 327, p. 42 

To draw a regular nonagon upon a 

given side .§ 33°, P- 43 

To draw a regular nonagon within a 

given circle § 318, p. 40 

To draw a regular octagon upon a given 

side § 329, P- 43 

To draw a regular octagon within a 

given circle § 3'7, P- 4° 

To draw a regular octagon within a 

given square § 33°> P- 45 

To draw a regular pentagon upon a 

given side § 326, P- 42 

To draw a regular pentagon within a 

given circle § 3'4. P- 39 

To draw a regular polygon of eleven 

sides within a given circle. . . § 323, p. 41 
To draw a regular undecagon upon a 

given side § 332, p. 43 

To draw a regular undecagon within a 

given circle § 320, p. 40 

To draw a scroll to a spocified width as 

for a bracket or modillion § 310, p. 38 

To draw a simple volute § 306, p. 36 



242 



Index. 



To draw a spiral by means of a spool and 

thread § 309, p. 38 

To draw a spiral from centers with com- 
passes § 30S, p. 38 

To draw a square about a given circle . . 

§ 354, P- 50 
To draw a square upon a given side. . . . 

§348, 355, PP- 48, 51 
To draw a square within a given circle. 

§ 313, 344, 361, PP- 39, 47, 52 

To draw a straight line equal to any part 

of a circle less than a semicircle 

§ 300-301, p. 35 

To draw a straight line equal to the cir- 
cumference of a given circle 

§ 296-297, pp. 34-35 

To draw a straight line equal to the 
quarter circumference of a given 
circle § 299, p. 35 

To draw a straight line equal tosemicir- 
cumference of a given circle. .§ 29S, p. 35 

To draw a straight line parallel to a 
given line and at a given distance 
from it, using the compasses and a 
straight-edge § 277, p. 29 

To draw a tangent to a circle or a por- 
tion of a circle without having re- 
course to the center § 295, p. 34 

To erect a perpendicular at a given 
point in a straight line by means of 
compasses and straight-edge. .§ 279, p. 30 

To erect a perpendicular at or near the 
end of a given straight line by means 
of the compasses and straight-edge . . 

§ 280-281, p. 30 

To establish the profile of a correspond- 
ing inclined molding, from a given 
horizontal molding, to miter with it at 
right angles in plan and the several 
miter patterns involved § 550, p. 162 

To find the center from which a given 
arc is struck § 289, p. 33 

To find the center from which a given 
arc may be struck, by the use of the 
square § 291, p. 33 

To find the centers aud true axis of an 
ellipse § 400, p. 61 

To find the centers in a given ellipse by 
which an approximate figure may be 
constructed § 401, p. 61 

To project a figure § 200, p. 14 

To rake a molding § 159, p. 12 

To stnke a segment of a circle by a 
triangular guide, the chord and bight 
being given 8 292, p. 34 

To raise § 203, p. 14 

To raise a perpendicular to an arc of a 
circle without having recourse to the 
center § 294, p. 34 

Tools and materials, Drawing p. 15 

Top, The pattern of an article having an 
elliptical base and a round. . . .§ 4S3, p. 96 

Top square, The pattern of a flaring 
article of which the base is an oblong 
and the § 471, p. 86 

Top, The pattern of an oblong flaring 
article having a § 482, p. 95 

Top of which is round and the bottom of 
which is oblong with semicircular 
ends, The pattern of a flaring article 
the §480, p. 93 

Top of the center being 1< cated near 
one end, The pattern of a flaring arti- 
cle the top of which is round and 
the bottom of which is oblong with 
semicircular ends, the § 484, p. 97 

Top, The pattern of a flaring article the 
base of w hich is a rectangle and top, of 
which is round, the center of the top 
being toward one end § 481, p. 94 

Tracing cloth § 275, p. 28 

Tracing paper § 275, p. 28 

Trammel, An improvised § 385, p. 57 



Trammel, The construction of a.§ 382, p. 56 
Trammel, To draw an ellipse to given 

dimensions by means of a § 384, p. 57 

Trammels, Beam compasses and 

§ 240-243, p. 21 
Transferring distances, Use of the T- 

square in § 439, p. 7 1 

Transition between an octagon and a 

square, the gore piece § 565, p. 176 

Transition from an octagon to a square, 
The gore piece in a molding forming. 

§ 56_4, P- 175 

Transverse axis § 84, p. 6 

Trapezium § 40, p. 3 

Trapezoid § 41, p. 3 

Tray, The pattern of a heart-shaped 

flaring § 479, p. 92 

Treatise on pattern cutting, A § 1, p. 1 

Triangle, A § 26-32, pp. 2-3 

Triangle, A right cone generated by the 
revolution of a right-angled. . .§ 441, p. 72 

Triangle prism, A §9', P- 7 

Triangle, To construct a, the length of 

three sides being given § 325, p. 41 

Triangle, To construct an equilateral, 

thelengthof a side being given. §365, p. 53 
Triangle, To construct an equilateral 

upon a given side § 324, p. 41 

Triangle, To draw an equilateral, upon 

a given side § 347, p. 48 

Triangle, To draw an equilateral, about 

a given circle § 351, p. 49 

Triangle, To draw an equilateral, within 

a given circle § 360, p. 51 

Triangle, The patterns for a pedestal of 

which the plan is (inequilateral. §585, p. 207 
Triangle, The patterns of the moldings 
upon a panel the shape of which is a 

scalene § 545, p. 158 

Triangle within a given circle, To draw 

an equilateral § 3(2, 343, pp. 39, 46 

Triangles for generating an elliptical 

cone 8 452, p. 76 

Triangles or set squares g 233, p. 19 

Triangles or set squares and the T-.-quaro 

in the constiuction of polygons P- 45 

Triangles, To construct a hexagon by 
means of six equilateral, the length of 

a side being given § 367, p. 54 

Triangles used in the development of the 
pattern of an elliptical cone, Dia- 
gram of §452, P- 77 

Triangular frustum obtained by revolu- 
tion, Covering of a § 448, p. 75 

Triangular guide for striking the seg- 
ment of a circle § 292, p. 34 

Triangular pyramid, The envelope of a. 

§461, p. 81 

TriaDgular scales $ 250, p. 23 

Triangular pyramids § 102, p. 8 

Truncated cone, A § 100, p. 8 

Truss, A § 139, p. II 

T-square in dropping points § 439, p. 71 

T-square in pattern cutting illustrated, 

Use of the § 439, p. 71 

T square in transferring distances, Use 

of the § 439, p. 71 

T-square, triangles or set squares, in the 

construction of polygons ,p, 45 

T-square, Use of, in miter cutting. § 418, p. 66 

T squares § 225-228, p. 18 

Tubs, Oval foot § 492, figs. 337-339, 

Tumblers § 488, fig. 333 

Two circles and intersecting lines, To 
construct an ellipse to given dimen- 
sions by the use of § 392, p. 59 

Two-piece elbow, A § 509, p. 121 

Two piece elbow in elliptical pipe, A . . . 

§ 520, p. 133 
Two-piece elbow in tapering pipe, A.. ... 

§ 5i6, P- 127 
Undecagon, To draw a regular, upon a 
given side § 332, p. 43 



Undecagon, To draw a regular within a 

given circle § 320, p. 40 

Ungula § 116, p. 9 

Uru, The pattern for an, the plan of 

which is a dodecagon § 593, p. 213 

Urn, The patterns for a vase or, in any 

number of pieces § 440, p. 72 

Use of lines in laying off the pattern of 

covering for molding § 412-413, p. 65 

Use of the T-square in dropping points.. 

„ , _ §439,P- 71 

Use of the T-square in miter cutting, 

The §418, p. 66 

Use of the T-square in pattern cutting 

illustrated § 439, p. 71 

Use of the T-square in transferring dis- 
tances § 439, p. 71 

Using the protractor § 358, p. 51 

Usual method for describing patterns of 

regular flaiing ware, The § 454, p. 77 

Vase or urn in any number of pieces, 

The patterns for a § 440, p. 72 

Vase, The patterns for an elliptical con- 
struction in twelve pieces. . . .§594, p. 214 
Vase, Patterns of an octagonal. § 439, p. 71 
Vase, The patterns for a, the plan of 

which is a pentagon § 587, p. 207 

Vase the plan of which is a heptagon, 

The patterns for a § 589, p. 2C9 

Ventilator, Patterns for a ship, having 
oval mouth on around pipe. .§ 605, p. 223 

Versed sine of an aro, The § 81, p 6 

Vertex §37, P- 3 

Vertex of a pyramid, The § 102, p. 8 

Vertical lines § 13, p. 2 

Vertical lines in a drawing, Method of 

indicating § 14, p. 2 

Vertical molding, A § 152, p. 11 

Vertical section, A • ■ • •§ 181, p. 12 

Vessel with round ends. The pattern of 

a raised cover fitting an oblong. § 476, p. 90 
Volute, To describe an Ionic. . . .§ 307, p. 37 

Volute, To draw a simple § 306, p. 36 

Volute of a capital, Patterns for the. . . 

§ 602, p. 220 
Ware, Composition of the shape of reg- 
ular flaring § 458, p. 80 

Ware corresponds, Analyses of (he solid 
to the envelope of which regular flar- 
ing §455, P- 78 

Ware corresponds, The string or thread 
method for developing the envelope 
of the solid to which regular flaring.. 

§457, P- 79 
Ware corresponds, The solid to the envel- 
ope of which regular flaring. .§ 453, p. 77 
Ware, The usual method for describing 

patterns of regular flaring. . . .§ 454, p. 77 
Wash at the bottom, The patterns of a 
hip molding upon an octagon angle of 
a mansard roof mitering against aD 

inclined § 577, p. 193 

Wash basins 8 488, fig. 333 

Wash boiler covers 8 475, fig- 311 

Wash bowls § 488, fig. 333 

Wash kettles § 488, fig. 333 

White drawing paper § 273, p. 27 

Window cap and its several patterns, A 

plain § 567, p. 17S 

Window cap, Patterns for simple curved 

moldings in a § 568, p. 179 

Window cap, The patterns for elliptical 

curved moldings in a § 569, p. 181 

Window cap, The patterns in a com- 
mon § 437, p. T> 

Wine cooler, Flaring § 488, fig. 333 

Wood, To draw an ellipse of given dimen- 
sions by means of a square and a strip 

of "'"•••■§ 386, p. 57 

Wooes for drawing tables § 210, p. 15 

Working drawing, A § 186, p. 13 

Zones, To construct a ball of pieces the 
general Bhape of § 598, p. 218 






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